diff --git a/Q3/Documentation/English/Index/_0.cfs b/Q3/Documentation/English/Index/_0.cfs index 248cc0891..bdab4309a 100644 Binary files a/Q3/Documentation/English/Index/_0.cfs and b/Q3/Documentation/English/Index/_0.cfs differ diff --git a/Q3/Documentation/English/Index/segments_3 b/Q3/Documentation/English/Index/segments_3 index 7a3a7508b..8f8ac6493 100644 Binary files a/Q3/Documentation/English/Index/segments_3 and b/Q3/Documentation/English/Index/segments_3 differ diff --git a/Q3/Documentation/English/ReferencePages/Symbols/BlockEncoding.nb b/Q3/Documentation/English/ReferencePages/Symbols/BlockEncoding.nb index aeeea594e..b5dbe3fa3 100644 --- a/Q3/Documentation/English/ReferencePages/Symbols/BlockEncoding.nb +++ b/Q3/Documentation/English/ReferencePages/Symbols/BlockEncoding.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 100668, 2427] -NotebookOptionsPosition[ 91706, 2246] -NotebookOutlinePosition[ 94116, 2305] -CellTagsIndexPosition[ 94030, 2300] +NotebookDataLength[ 171619, 4097] +NotebookOptionsPosition[ 158193, 3828] +NotebookOutlinePosition[ 160667, 3889] +CellTagsIndexPosition[ 160580, 3884] WindowTitle->BlockEncoding WindowFrame->Normal*) @@ -27,11 +27,11 @@ Cell[BoxData[GridBox[{ TemplateBox[{12}, "Spacer1"], Cell["Q3 SYMBOL", "PacletNameCell", TextAlignment->Center,ExpressionUUID-> - "05d01478-8e35-411f-a93e-2ed77416cc4e"], + "5b368875-50e4-4a42-aa14-a451a68ee219"], TemplateBox[{8}, "Spacer1"]}]], TextAlignment->Center,ExpressionUUID-> - "f64e5770-86d8-4f3a-a570-edb06a0856e6"], + "941a989a-6ea3-423b-9b7a-5978fe8c42d8"], Background->RGBColor[0.490196, 0.576471, 0.690196], ItemSize->Full], ""} }, @@ -54,8 +54,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "629c851e-b7c8-46d8-b5bf-319af85192db"] - }],ExpressionUUID->"bc9b8be0-9793-4217-8ef7-9211061bd09c"], + "f47feb8a-a775-4d35-b60a-526d54bbcab6"] + }],ExpressionUUID->"cf01a21b-9056-4c2b-b3a7-64600213c61e"], StripOnInput->False],{ "\"UniformlyControlledGate\"" :> Documentation`HelpLookup["paclet:Q3/ref/UniformlyControlledGate"]}, @@ -64,7 +64,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"SeeAlso"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "637f546d-7993-4839-a57a-9d11af16e63a"], + "c244003e-8ac9-4bc8-8c16-995e44a9c7c3"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -83,8 +83,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "78de9b67-5a10-4040-a8b1-5e397356b981"] - }],ExpressionUUID->"7c664053-e998-440f-a226-e477280b16b5"], + "d0fcf528-68d5-4331-ad90-9990f807facc"] + }],ExpressionUUID->"368db603-7925-40ce-b77c-b3ad1b3e3a52"], StripOnInput->False],{ "\"Quantum Information Systems\"" :> Documentation`HelpLookup[ @@ -95,7 +95,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"MoreAbout"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "3346427c-ab00-42f1-af0c-a09b33736854"], + "59475cba-60e6-467e-bd54-3d5598c13f96"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -114,8 +114,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "f14e6a07-d88e-4c48-9f56-6f41162178e7"] - }],ExpressionUUID->"edda4eab-bf81-4df2-8236-79ae52fbfd23"], + "c4957aeb-05b2-4f04-b7a3-98b61aa57599"] + }],ExpressionUUID->"b59efeb4-811b-4630-9e0d-8ea90deaceeb"], StripOnInput->False],{ "\"Block Encoding\"" :> Documentation`HelpLookup["paclet:Q3/tutorial/BlockEncoding"], @@ -132,7 +132,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"Tutorials"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "8e4d25b6-2a95-4494-9a49-7d5832f1415a"], + "40b3b18a-9baa-4129-a886-0825a822fb96"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -151,8 +151,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "77967377-aa30-438a-86d0-092d00fef4f9"] - }],ExpressionUUID->"19cb6fed-59bc-4730-9bca-27088d183b4b"], + "fcf3a009-54fd-40f4-b399-b190515f4cec"] + }],ExpressionUUID->"c870cb2a-55c5-4cda-b7e4-8d7935dec381"], StripOnInput->False],{ "\"Q3/ref/BlockEncoding\"" :> None, "\"Copy Wolfram Documentation Center URL\"" :> @@ -189,14 +189,14 @@ BlockEncoding.html"], StandardForm]], "Input", TextClipboardType -> MenuStyle->"URLMenu"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "6cd9b175-43e6-43e6-8dd4-c945c6b265e5"] + "e32d51bc-1f29-41c6-b5cd-d9d71e1565f2"] }], "AnchorBar", CacheGraphics->False,ExpressionUUID-> - "e5d52520-f666-427b-89f2-32929d8f8cd1"]} + "0b49bea4-27b4-46b3-aa00-bac93817a85c"]} }]], "AnchorBarGrid", - CellID->1,ExpressionUUID->"3ad59a88-d0b0-42bb-8241-a66d5e67645c"], + CellID->1,ExpressionUUID->"c60dd3cf-c4e4-4402-aa00-20b047b6a66e"], -Cell["Q3`", "ContextNameCell",ExpressionUUID->"5dab263b-ef9b-44cd-9010-d921d054167e"], +Cell["Q3`", "ContextNameCell",ExpressionUUID->"569d58b3-cd84-4fee-b3e9-c20aa4887062"], Cell[CellGroupData[{ @@ -204,14 +204,14 @@ Cell[BoxData[GridBox[{ {Cell[TextData[{ Cell[ "BlockEncoding", "ObjectName",ExpressionUUID-> - "19d3f0d3-fb11-451e-b6f1-ceefc21cb255"], + "f68eea09-a088-4d03-ab11-5a9bbc9db9f5"], Cell[BoxData[ TemplateBox[{8}, - "Spacer1"]],ExpressionUUID->"fd81a412-56c9-4a7b-9022-b8f1a6797ed3"], + "Spacer1"]],ExpressionUUID->"bacf6088-36cd-4deb-9596-5221a610e07e"], Cell[BoxData[ ""], "ObjectNameTranslation",ExpressionUUID-> - "aef7968c-d2a5-4fca-b777-6e7a143d0796"] - }],ExpressionUUID->"e04442b1-249c-40c0-beaa-a5ddd23d074b"], Cell[BoxData[ + "cdb5d67d-aa31-4679-9c47-b0096d015179"] + }],ExpressionUUID->"b6723cb3-24d5-4f8f-82f9-9520f2367b9b"], Cell[BoxData[ TooltipBox[ StyleBox[ TagBox[ @@ -236,9 +236,9 @@ Cell[BoxData[GridBox[{ "New in 14", TooltipDelay->0.3]], Magnification->1,ExpressionUUID-> - "7e4285d1-f7d0-486f-82b7-e700200f700e"]} + "31c16c5b-d1d5-418b-b7f7-20befb4b19a3"]} }]], "ObjectNameGrid", - CacheGraphics->False,ExpressionUUID->"783804fb-3881-4a8d-b8a3-1056bbd1a7e4"], + CacheGraphics->False,ExpressionUUID->"2ce97ba2-849d-402a-bc91-0c9a8d3b6798"], Cell[BoxData[GridBox[{ {"", Cell[TextData[{ @@ -303,7 +303,7 @@ Cell[BoxData[GridBox[{ FontFamily->"Source Sans Pro",ExpressionUUID-> "b18f167f-3474-4512-b9aa-a696d174b546"], "." - }],ExpressionUUID->"54c0c458-e336-4a48-b416-3df5c7d103bb"]} + }],ExpressionUUID->"0da14748-4dff-45ae-81ed-1548916285ea"]} }]], "Usage", CellID->2139940575,ExpressionUUID->"8413d501-f1c5-494e-8418-ab42652b7777"] }, Open ]], @@ -336,12 +336,12 @@ Cell[TextData[Cell[BoxData[ PlotRange -> {{-3, 4}, {-1, 1}}, ImageSize -> 20], Magnification -> 0.68 Inherited], Rational[1, 2] Pi, {-1.65, -1}]]], ImageSizeCache->{14., {0.45000000000000107`, 13.549999999999999`}}]], - ExpressionUUID->"730fd38f-68d1-4566-a905-7e95ba1bf74a"], + ExpressionUUID->"a70823a6-8b73-4062-a6aa-48c61249dc48"], Cell[BoxData[ TemplateBox[{1}, - "Spacer1"]],ExpressionUUID->"93b78832-57bb-4260-a3e3-0365a367e065"], + "Spacer1"]],ExpressionUUID->"f644c8d1-e5b6-4bed-8db0-2d12f366c9ba"], "Details and Options" - }], "NotesFrameText",ExpressionUUID->"f3722c6d-3f15-4308-ac9b-4d9c05898cc8"], + }], "NotesFrameText",ExpressionUUID->"502ef95b-9b1d-4b83-b898-bf241ae6dd0b"], Appearance->{Automatic, None, "Normal", Automatic}, BaseStyle->None, ButtonFunction:>(FrontEndExecute[{ @@ -352,12 +352,12 @@ Cell[TextData[Cell[BoxData[ FrontEnd`SelectedNotebook[], After, CellContents]}]& ), Evaluator->None, Method-> - "Preemptive"]],ExpressionUUID->"ef7866c7-c03b-4892-8c1f-78437059e011"]], \ + "Preemptive"]],ExpressionUUID->"34363de8-97f7-47d5-92d4-6e5f04f5fba7"]], \ "NotesSection", WholeCellGroupOpener->True, CellGroupingRules->{"SectionGrouping", 50}, CacheGraphics->False, - CellID->1192351708,ExpressionUUID->"c25df7f4-ecb6-4598-9c01-ef5756a6b295"], + CellID->203830331,ExpressionUUID->"6b1a341f-bac2-47b6-aeed-1d8b128dd65b"], Cell[TextData[{ "Q3 implements one particular form described in the tech note \"", @@ -366,7 +366,117 @@ Cell[TextData[{ ButtonData->"paclet:Q3/tutorial/BlockEncoding"], "\"." }], "Notes", - CellID->2072607716,ExpressionUUID->"a1ecd3b3-5c7b-445b-89e3-919f08c38308"] + CellID->2072607716,ExpressionUUID->"a1ecd3b3-5c7b-445b-89e3-919f08c38308"], + +Cell[TextData[{ + "Option ", + Cell[BoxData[ + RowBox[{"\"\\"", "\[Rule]", + RowBox[{"{", + RowBox[{ + SubscriptBox[ + StyleBox["lbl", "TI"], + StyleBox["1", "TR"]], ",", + RowBox[{"{", + RowBox[{ + SubscriptBox[ + StyleBox["lbl", "TI"], + StyleBox["21", "TR"]], ",", + SubscriptBox[ + StyleBox["lbl", "TI"], + StyleBox["22", "TR"]], ",", + SubscriptBox[ + StyleBox["lbl", "TI"], + StyleBox["23", "TR"]]}], "}"}]}], "}"}]}]], "InlineFormula", + FontFamily->"Source Sans Pro",ExpressionUUID-> + "84c6897a-9912-466c-9b3d-9e1e0dadf511"], + " sets the label of the pristine form of ", + Cell[BoxData[ + TemplateBox[{ + Cell[ + TextData["BlockEncoding"]], "paclet:Q3/ref/BlockEncoding", + "Q3 Package Symbol"}, + "PackageLink", + BaseStyle->"InlineFormula"]], "InlineFormula", + FontFamily->"Source Sans Pro",ExpressionUUID-> + "1dd41123-91ae-48ec-bd99-365d3a6eb2e8"], + " to ", + Cell[BoxData[ + SubscriptBox[ + StyleBox["lbl", "TI"], + StyleBox["1", "TR"]]], "InlineFormula", + FontFamily->"Source Sans Pro",ExpressionUUID-> + "5c6e1e3c-a1c7-4cc5-aa84-b438f9683f8f"], + ", and ", + Cell[BoxData[ + RowBox[{"{", + RowBox[{ + SubscriptBox[ + StyleBox["lbl", "TI"], + StyleBox["21", "TR"]], ",", + SubscriptBox[ + StyleBox["lbl", "TI"], + StyleBox["22", "TR"]], ",", + SubscriptBox[ + StyleBox["lbl", "TI"], + StyleBox["22", "TR"]]}], "}"}]], "InlineFormula", + FontFamily->"Source Sans Pro",ExpressionUUID-> + "5ef9aeef-440e-414e-9c0d-3d80328730b2"], + " for the labels of the expanded form (", + Cell[BoxData[ + RowBox[{ + TemplateBox[{ + Cell[ + TextData["Expand"]], "paclet:ref/Expand"}, + "RefLink", + BaseStyle->{"InlineFormula"}], "[", + RowBox[{"BlockEncoding", "[", + StyleBox["\[Ellipsis]", "TR"], "]"}], "]"}]], "InlineFormula", + FontFamily->"Source Sans Pro",ExpressionUUID-> + "dcaba06b-4a14-4f47-8213-cd664e263483"], + ")." +}], "Notes", + CellID->1333943237,ExpressionUUID->"c42c3b63-4aac-47b3-aa3c-d4d2c09f8110"], + +Cell[TextData[{ + "Option ", + Cell[BoxData[ + RowBox[{"\"\\"", "\[Rule]", + StyleBox["lbl", "TI"]}]], "InlineFormula", + FontFamily->"Source Sans Pro",ExpressionUUID-> + "ece4b4ec-431e-44bf-bb13-7e56f9199d91"], + " or ", + Cell[BoxData[ + RowBox[{"\"\\"", "\[Rule]", + RowBox[{"{", + StyleBox["lbl", "TI"], "}"}]}]], "InlineFormula", + FontFamily->"Source Sans Pro",ExpressionUUID-> + "7e720e9b-d3f2-476e-b7a2-7b8a543cc3b1"], + " just sets the label of the pristine form of ", + Cell[BoxData[ + TemplateBox[{ + Cell[ + TextData["BlockEncoding"]], "paclet:Q3/ref/BlockEncoding", + "Q3 Package Symbol"}, + "PackageLink", + BaseStyle->"InlineFormula"]], "InlineFormula", + FontFamily->"Source Sans Pro",ExpressionUUID-> + "22493e90-b00f-453c-9aba-a63b47bea5d6"], + "; the expanded form (", + Cell[BoxData[ + RowBox[{ + TemplateBox[{ + Cell[ + TextData["Expand"]], "paclet:ref/Expand"}, + "RefLink", + BaseStyle->{"InlineFormula"}], "[", + RowBox[{"BlockEncoding", "[", + StyleBox["\[Ellipsis]", "TR"], "]"}], "]"}]], "InlineFormula", + FontFamily->"Source Sans Pro",ExpressionUUID-> + "13fe7ac8-f6ee-425b-b768-a46044199ee4"], + ") takes the default label." +}], "Notes", + CellID->320383127,ExpressionUUID->"ba388e21-3c75-489d-a0de-1929a4e34416"] }, Dynamic[CurrentValue[ EvaluationNotebook[], {TaggingRules, "Openers", "NotesSection"}, Closed]]]], @@ -397,19 +507,19 @@ Cell[TextData[{ PlotRange -> {{-3, 4}, {-1, 1}}, ImageSize -> 20], Magnification -> 0.68 Inherited], Rational[1, 2] Pi, {-1.65, -1}]]], ImageSizeCache->{14., {5., 9.}}]],ExpressionUUID-> - 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CellID->1478085105,ExpressionUUID->"75c62446-dbb1-4773-a4ce-464a2d8487f2"] + CellID->1224459765,ExpressionUUID->"839dd08c-2bd5-45ba-bf83-b04f3b0b4c89"] }, Open ]], Cell["\<\ @@ -650,7 +760,7 @@ Cell[BoxData[ 0, Rational[-1, 8]], Complex[0, Rational[-1, 4]], Complex[ 0, Rational[1, 8]], Complex[0, Rational[-1, 4]]}}]]]]], "Output", CellLabel->"Out[5]//MatrixForm=", - CellID->388766686,ExpressionUUID->"141085d5-5033-4d1a-bcf9-5698f344995e"] + CellID->230027356,ExpressionUUID->"71e314a4-5f75-41a2-acee-6c1d8e94dfec"] }, Open ]], Cell["Block-encode the matrix using the ancilla register.", "ExampleText", @@ -821,7 +931,7 @@ Cell[BoxData[ SelectWithContents->True, Selectable->False]], "Output", CellLabel->"Out[6]=", - CellID->394892105,ExpressionUUID->"89a44ab0-c820-4aa4-aa49-2cf6b5c5932e"] + CellID->375592456,ExpressionUUID->"897686bd-220f-42f7-9c29-63286a940198"] }, Open ]], Cell["Display the block encoding in a quantum circuit.", "ExampleText", @@ -888,7 +998,7 @@ Cell[BoxData[ $CellContext`A[1, Q3`$], $CellContext`A[2, Q3`$]}]]]], "Output", CellLabel->"Out[7]=", - CellID->781168981,ExpressionUUID->"9be34463-f2e2-4d0a-88f2-ad8c7103ce30"] + CellID->902256453,ExpressionUUID->"b462011b-5f1b-489d-a8fd-8df4a0ae74ee"] }, Open ]], Cell["\<\ @@ -902,7 +1012,7 @@ Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"mat", "=", - RowBox[{"Matrix", "[", "op", "]"}]}], ";"}], "\[IndentingNewLine]", + RowBox[{"Matrix", "[", "enc", "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"mat", "[", RowBox[{"[", @@ -972,7 +1082,7 @@ Cell[BoxData[ 0, Rational[-1, 4]], Complex[0, Rational[-1, 8]], Complex[ 0, Rational[-1, 4]], Complex[0, Rational[1, 8]]}}]]]]], "Output", CellLabel->"Out[8]//MatrixForm=", - CellID->2094726860,ExpressionUUID->"5528ed8c-d449-4c7d-9575-02a0fbe14a24"] + CellID->330995392,ExpressionUUID->"8c4940af-782f-4e12-8329-0c2793475561"] }, Open ]], Cell[TextData[{ @@ -1073,7 +1183,7 @@ Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"new", "=", - RowBox[{"Expand", "[", "op", "]"}]}], "\[IndentingNewLine]", + RowBox[{"Expand", "[", "enc", "]"}]}], "\[IndentingNewLine]", RowBox[{"alt", "=", RowBox[{"Expand", "[", "new", "]"}]}]}], "Input", CellProlog:>Needs["Q3`"], @@ -1237,7 +1347,7 @@ Cell[BoxData[ $CellContext`A[1, Q3`$], $CellContext`A[2, Q3`$]}]]], "Output", CellLabel->"Out[9]=", - 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16] Pi]] + + Rational[3, 4] E^(2 Cos[Rational[1, 16] Pi]) Cos[Rational[1, 16] Pi]^2 + Cos[Rational[3, 16] Pi] Sin[2 Pi Cos[Rational[1, 16] Pi]] + + Rational[-3, 4] E^((-2) Cos[Rational[3, 16] Pi]) + Cos[Rational[3, 16] Pi]^2 Sin[Rational[1, 16] Pi] + Sin[2 Pi Cos[Rational[3, 16] Pi]] + + Rational[-3, 4] E^(2 Cos[Rational[3, 16] Pi]) Cos[Rational[3, 16] Pi]^2 + Sin[Rational[1, 16] Pi] Sin[2 Pi Cos[Rational[3, 16] Pi]] + + Rational[-3, 4] E^((-2) Sin[Rational[1, 16] Pi]) + Sin[Rational[1, 16] Pi]^2 Sin[Rational[3, 16] Pi] + Sin[2 Pi Sin[Rational[1, 16] Pi]] + + Rational[-3, 4] E^(2 Sin[Rational[1, 16] Pi]) Sin[Rational[1, 16] Pi]^2 + Sin[Rational[3, 16] Pi] Sin[2 Pi Sin[Rational[1, 16] Pi]] + + Rational[-3, 4] E^((-2) Sin[Rational[3, 16] Pi]) Cos[Rational[1, 16] Pi] + Sin[Rational[3, 16] Pi]^2 Sin[2 Pi Sin[Rational[3, 16] Pi]] + + Rational[-3, 4] E^(2 Sin[Rational[3, 16] Pi]) Cos[Rational[1, 16] Pi] + Sin[Rational[3, 16] Pi]^2 Sin[2 Pi Sin[Rational[3, 16] Pi]], + Rational[3, 4] 2^Rational[-1, 2] E^((-2) Cos[Rational[1, 16] Pi]) + Cos[Rational[1, 16] Pi]^2 Sin[2 Pi Cos[Rational[1, 16] Pi]] + + Rational[-3, 4] 2^Rational[-1, 2] E^(2 Cos[Rational[1, 16] Pi]) + Cos[Rational[1, 16] Pi]^2 Sin[2 Pi Cos[Rational[1, 16] Pi]] + + Rational[-3, 4] 2^Rational[-1, 2] E^((-2) Cos[Rational[3, 16] Pi]) + Cos[Rational[3, 16] Pi]^2 Sin[2 Pi Cos[Rational[3, 16] Pi]] + + Rational[3, 4] 2^Rational[-1, 2] E^(2 Cos[Rational[3, 16] Pi]) + Cos[Rational[3, 16] Pi]^2 Sin[2 Pi Cos[Rational[3, 16] Pi]] + + Rational[3, 4] 2^Rational[-1, 2] E^((-2) Sin[Rational[1, 16] Pi]) + Sin[Rational[1, 16] Pi]^2 Sin[2 Pi Sin[Rational[1, 16] Pi]] + + Rational[-3, 4] 2^Rational[-1, 2] E^(2 Sin[Rational[1, 16] Pi]) + Sin[Rational[1, 16] Pi]^2 Sin[2 Pi Sin[Rational[1, 16] Pi]] + + Rational[-3, 4] 2^Rational[-1, 2] E^((-2) Sin[Rational[3, 16] Pi]) + Sin[Rational[3, 16] Pi]^2 Sin[2 Pi Sin[Rational[3, 16] Pi]] + + Rational[3, 4] 2^Rational[-1, 2] E^(2 Sin[Rational[3, 16] Pi]) + Sin[Rational[3, 16] Pi]^2 Sin[2 Pi Sin[Rational[3, 16] Pi]], + Rational[3, 4] E^((-2) Cos[Rational[1, 16] Pi]) Cos[Rational[1, 16] Pi]^2 + Sin[Rational[3, 16] Pi] Sin[2 Pi Cos[Rational[1, 16] Pi]] + + Rational[3, 4] E^(2 Cos[Rational[1, 16] Pi]) Cos[Rational[1, 16] Pi]^2 + Sin[Rational[3, 16] Pi] Sin[2 Pi Cos[Rational[1, 16] Pi]] + + Rational[-3, 4] E^((-2) Cos[Rational[3, 16] Pi]) Cos[Rational[1, 16] Pi] + Cos[Rational[3, 16] Pi]^2 Sin[2 Pi Cos[Rational[3, 16] Pi]] + + Rational[-3, 4] E^(2 Cos[Rational[3, 16] Pi]) Cos[Rational[1, 16] Pi] + Cos[Rational[3, 16] Pi]^2 Sin[2 Pi Cos[Rational[3, 16] Pi]] + + Rational[3, 4] E^((-2) Sin[Rational[1, 16] Pi]) Cos[Rational[3, 16] Pi] + Sin[Rational[1, 16] Pi]^2 Sin[2 Pi Sin[Rational[1, 16] Pi]] + + Rational[3, 4] E^(2 Sin[Rational[1, 16] Pi]) Cos[Rational[3, 16] Pi] + Sin[Rational[1, 16] Pi]^2 Sin[2 Pi Sin[Rational[1, 16] Pi]] + + Rational[3, 4] E^((-2) Sin[Rational[3, 16] Pi]) Sin[Rational[1, 16] Pi] + Sin[Rational[3, 16] Pi]^2 Sin[2 Pi Sin[Rational[3, 16] Pi]] + + Rational[3, 4] E^(2 Sin[Rational[3, 16] Pi]) Sin[Rational[1, 16] Pi] + Sin[Rational[3, 16] Pi]^2 Sin[2 Pi Sin[Rational[3, 16] Pi]], + Rational[3, 4] E^((-2) Cos[Rational[1, 16] Pi]) Cos[Rational[1, 16] Pi]^2 + Sin[Rational[1, 8] Pi] Sin[2 Pi Cos[Rational[1, 16] Pi]] + + Rational[-3, 4] E^(2 Cos[Rational[1, 16] Pi]) Cos[Rational[1, 16] Pi]^2 + Sin[Rational[1, 8] Pi] Sin[2 Pi Cos[Rational[1, 16] Pi]] + + Rational[-3, 4] E^((-2) Cos[Rational[3, 16] Pi]) Cos[Rational[1, 8] Pi] + Cos[Rational[3, 16] Pi]^2 Sin[2 Pi Cos[Rational[3, 16] Pi]] + + Rational[3, 4] E^(2 Cos[Rational[3, 16] Pi]) Cos[Rational[1, 8] Pi] + Cos[Rational[3, 16] Pi]^2 Sin[2 Pi Cos[Rational[3, 16] Pi]] + + Rational[-3, 4] E^((-2) Sin[Rational[1, 16] Pi]) + Sin[Rational[1, 16] Pi]^2 Sin[Rational[1, 8] Pi] + Sin[2 Pi Sin[Rational[1, 16] Pi]] + + Rational[3, 4] E^(2 Sin[Rational[1, 16] Pi]) Sin[Rational[1, 16] Pi]^2 + Sin[Rational[1, 8] Pi] Sin[2 Pi Sin[Rational[1, 16] Pi]] + + Rational[3, 4] E^((-2) Sin[Rational[3, 16] Pi]) Cos[Rational[1, 8] Pi] + Sin[Rational[3, 16] Pi]^2 Sin[2 Pi Sin[Rational[3, 16] Pi]] + + Rational[-3, 4] E^(2 Sin[Rational[3, 16] Pi]) Cos[Rational[1, 8] Pi] + Sin[Rational[3, 16] Pi]^2 Sin[2 Pi Sin[Rational[3, 16] Pi]], + Rational[3, 4] E^((-2) Cos[Rational[1, 16] Pi]) Cos[Rational[1, 16] Pi]^2 + Sin[Rational[1, 16] Pi] Sin[2 Pi Cos[Rational[1, 16] Pi]] + + Rational[3, 4] E^(2 Cos[Rational[1, 16] Pi]) Cos[Rational[1, 16] Pi]^2 + Sin[Rational[1, 16] Pi] Sin[2 Pi Cos[Rational[1, 16] Pi]] + + Rational[-3, 4] E^((-2) Cos[Rational[3, 16] Pi]) + Cos[Rational[3, 16] Pi]^2 Sin[Rational[3, 16] Pi] + Sin[2 Pi Cos[Rational[3, 16] Pi]] + + Rational[-3, 4] E^(2 Cos[Rational[3, 16] Pi]) Cos[Rational[3, 16] Pi]^2 + Sin[Rational[3, 16] Pi] Sin[2 Pi Cos[Rational[3, 16] Pi]] + + Rational[-3, 4] E^((-2) Sin[Rational[1, 16] Pi]) Cos[Rational[1, 16] Pi] + Sin[Rational[1, 16] Pi]^2 Sin[2 Pi Sin[Rational[1, 16] Pi]] + + Rational[-3, 4] E^(2 Sin[Rational[1, 16] Pi]) Cos[Rational[1, 16] Pi] + Sin[Rational[1, 16] Pi]^2 Sin[2 Pi Sin[Rational[1, 16] Pi]] + + Rational[3, 4] E^((-2) Sin[Rational[3, 16] Pi]) Cos[Rational[3, 16] Pi] + Sin[Rational[3, 16] Pi]^2 Sin[2 Pi Sin[Rational[3, 16] Pi]] + + Rational[3, 4] E^(2 Sin[Rational[3, 16] Pi]) Cos[Rational[3, 16] Pi] + Sin[Rational[3, 16] Pi]^2 Sin[2 Pi Sin[Rational[3, 16] Pi]]}, 0], + Editable->False, + SelectWithContents->True, + Selectable->False]], "Output", + CellLabel->"Out[5]=", + CellID->118276806,ExpressionUUID->"5291850e-1081-4ece-b920-7608be1ae157"] +}, Open ]], + +Cell["\<\ +Compare the approximate polynomial with the original function.\ +\>", "ExampleText", + CellID->436466161,ExpressionUUID->"8447f624-4961-41fb-86a9-34c623284540"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"Plot", "[", + RowBox[{ + RowBox[{"{", + RowBox[{ + RowBox[{"f", "[", "x", "]"}], ",", + RowBox[{"g", "[", "x", "]"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{"x", ",", + RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", "\[IndentingNewLine]", + 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"070a8371-ce8f-4f5a-80f3-a1b831ac7717"], + "dcce123f-3ddc-43e6-88fe-8cee14af74c4"], TemplateBox[{8}, "Spacer1"]}]], TextAlignment->Center,ExpressionUUID-> - "bcba44cf-0adc-48f1-b891-fb250480d1c7"], + "5f40a67f-0e2c-444c-8f98-c5f58372fc9b"], Background->RGBColor[0.490196, 0.576471, 0.690196], ItemSize->Full], ""} }, @@ -54,8 +54,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "d157deab-44c2-41b3-9d7b-b2316a56e402"] - }],ExpressionUUID->"fe2b7deb-c3bc-4719-9fc1-6711204b4ddd"], + "71d9c80f-0590-4798-9798-a7a1e953040b"] + }],ExpressionUUID->"3f7fd1ec-fe43-4583-bca2-06833f07b073"], StripOnInput->False],{ "\"ControlledGate\"" :> Documentation`HelpLookup["paclet:Q3/ref/ControlledGate"], "\"CNOT\"" :> @@ -68,7 +68,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"SeeAlso"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "35cefde9-324e-41ad-b580-5a08bf73e79d"], + "9bba5649-1789-4bb3-842c-77335a8df1e2"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -87,8 +87,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - 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SubscriptBox["1", - InterpretationBox[ - SubscriptBox[ - TemplateBox[{"S"}, "RowDefault"], - TemplateBox[{"1"}, "RowDefault"]], - $CellContext`S[1, Q3`$]]], + TemplateBox[{",", "\",\"", + InterpretationBox[ + SubscriptBox["S", + TemplateBox[{",", "\",\"", "1"}, "RowWithSeparators"]], + $CellContext`S[1, Q3`$]]}, "RowWithSeparators"]], SubscriptBox["1", - InterpretationBox[ - SubscriptBox[ - TemplateBox[{"S"}, "RowDefault"], - TemplateBox[{"2"}, "RowDefault"]], - $CellContext`S[2, Q3`$]]]}, "RowDefault"]}, + TemplateBox[{",", "\",\"", + InterpretationBox[ + SubscriptBox["S", + TemplateBox[{",", "\",\"", "2"}, "RowWithSeparators"]], + $CellContext`S[2, Q3`$]]}, "RowWithSeparators"]]}, + "RowWithSeparators"]}, "Ket"], Ket[<|$CellContext`S[1, Q3`$] -> 1, $CellContext`S[2, Q3`$] -> 1|>]], SqrtBox["2"]]}]}, @@ -2259,38 +2336,40 @@ Cell[BoxData[ FractionBox[ InterpretationBox[ TemplateBox[{ - TemplateBox[{ + TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", SubscriptBox["0", - InterpretationBox[ - SubscriptBox[ - TemplateBox[{"S"}, "RowDefault"], - TemplateBox[{"1"}, "RowDefault"]], - $CellContext`S[1, Q3`$]]], + TemplateBox[{",", "\",\"", + InterpretationBox[ + SubscriptBox["S", + TemplateBox[{",", "\",\"", "1"}, "RowWithSeparators"]], + $CellContext`S[1, Q3`$]]}, "RowWithSeparators"]], SubscriptBox["0", - InterpretationBox[ - SubscriptBox[ - TemplateBox[{"S"}, "RowDefault"], - TemplateBox[{"2"}, "RowDefault"]], - $CellContext`S[2, Q3`$]]]}, "RowDefault"]}, + TemplateBox[{",", "\",\"", + InterpretationBox[ + SubscriptBox["S", + TemplateBox[{",", "\",\"", "2"}, "RowWithSeparators"]], + $CellContext`S[2, Q3`$]]}, "RowWithSeparators"]]}, + "RowWithSeparators"]}, "Ket"], Ket[<|$CellContext`S[1, Q3`$] -> 0, $CellContext`S[2, Q3`$] -> 0|>]], SqrtBox["2"]], "+", FractionBox[ InterpretationBox[ TemplateBox[{ - TemplateBox[{ + TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", SubscriptBox["1", - InterpretationBox[ - SubscriptBox[ - TemplateBox[{"S"}, 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CellID->490037578,ExpressionUUID->"10e30fa3-7c9e-4614-93ec-8bc8c4ec6789"] }, Open ]], Cell["\<\ @@ -2322,34 +2401,32 @@ Cell[BoxData[ RowBox[{"mm", "=", RowBox[{"Measurements", "[", "qc", "]"}]}]], "Input", CellProlog:>Needs["Q3`"], - CellLabel->"In[7]:=", + CellLabel->"In[8]:=", CellID->1305283679,ExpressionUUID->"e0684c73-5d7d-4331-b831-461bc5f17408"], Cell[BoxData[ RowBox[{"{", InterpretationBox[ TemplateBox[{ - TemplateBox[{ + TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", InterpretationBox[ - TemplateBox[{ - TemplateBox[{"S"}, "RowDefault"], - TemplateBox[{"1"}, "RowDefault"], - TemplateBox[{"\"x\""}, "RowDefault"]}, "Subsuperscript", SyntaxForm -> - SubsuperscriptBox], + TemplateBox[{"S", + TemplateBox[{",", "\",\"", "1"}, "RowWithSeparators"], + TemplateBox[{",", "\",\"", "\"X\""}, "RowWithSeparators"]}, + "Subsuperscript", SyntaxForm -> SubsuperscriptBox], $CellContext`S[1, 1]], InterpretationBox[ - TemplateBox[{ - TemplateBox[{"S"}, "RowDefault"], - TemplateBox[{"2"}, "RowDefault"], - TemplateBox[{"\"x\""}, "RowDefault"]}, "Subsuperscript", SyntaxForm -> - SubsuperscriptBox], - $CellContext`S[2, 1]]}, "RowDefault"]}, + TemplateBox[{"S", + TemplateBox[{",", "\",\"", "2"}, "RowWithSeparators"], + TemplateBox[{",", "\",\"", "\"X\""}, "RowWithSeparators"]}, + "Subsuperscript", SyntaxForm -> SubsuperscriptBox], + $CellContext`S[2, 1]]}, "RowWithSeparators"]}, "RowDefault"], Q3`Multiply[ $CellContext`S[1, 1], $CellContext`S[2, 1]]], "}"}]], "Output", - CellLabel->"Out[7]=", - CellID->1171048129,ExpressionUUID->"e9b86038-f39b-44d4-80d6-68011a6f6972"] + CellLabel->"Out[8]=", + CellID->988440539,ExpressionUUID->"efe33073-53b5-4782-b862-2601b2c4e7f8"] }, Open ]] }, Open ]], @@ -2362,6 +2439,13 @@ Cell[BoxData[ $Line = 0; Null]], "ExampleDelimiter", CellID->585197025,ExpressionUUID->"4f438ce4-2f1e-462f-a8dd-6a3a42df61d7"], +Cell[BoxData[ + RowBox[{ + RowBox[{"SeedRandom", "[", "345", "]"}], ";"}]], "Input", + CellProlog:>Needs["Q3`"], + CellLabel->"In[1]:=", + CellID->1810676622,ExpressionUUID->"a3444aca-3e98-49ca-9790-07a062d0ff15"], + Cell["\<\ Here is another example with a vector with numeric coefficients.\ \>", "ExampleText", @@ -2371,7 +2455,7 @@ Cell[BoxData[ RowBox[{"Let", "[", RowBox[{"Qubit", ",", "S"}], "]"}]], "Input", CellProlog:>Needs["Q3`"], - CellLabel->"In[1]:=", + CellLabel->"In[2]:=", CellID->2028436802,ExpressionUUID->"68fa8f35-c7c5-4f7e-9a04-5978309bd579"], Cell[CellGroupData[{ @@ -2396,7 +2480,7 @@ Cell[BoxData[ RowBox[{"2", ",", "3", ",", "5"}], "}"}], "]"}], "\[Rule]", "1"}], "]"}]}]}], ")"}], "/", "3"}]}]], "Input", CellProlog:>Needs["Q3`"], - CellLabel->"In[2]:=", + CellLabel->"In[3]:=", CellID->572766723,ExpressionUUID->"e7a65c46-6d4a-45ac-9172-c12c3585136f"], Cell[BoxData[ @@ -2406,19 +2490,20 @@ Cell[BoxData[ RowBox[{ InterpretationBox[ TemplateBox[{ - TemplateBox[{ + TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", SubscriptBox["1", - InterpretationBox[ - SubscriptBox[ - TemplateBox[{"S"}, "RowDefault"], - TemplateBox[{"1"}, "RowDefault"]], - $CellContext`S[1, Q3`$]]], + TemplateBox[{",", "\",\"", + InterpretationBox[ + SubscriptBox["S", + TemplateBox[{",", "\",\"", "1"}, "RowWithSeparators"]], + $CellContext`S[1, Q3`$]]}, "RowWithSeparators"]], SubscriptBox["1", - InterpretationBox[ - SubscriptBox[ - TemplateBox[{"S"}, "RowDefault"], - TemplateBox[{"3"}, "RowDefault"]], - $CellContext`S[3, Q3`$]]]}, "RowDefault"]}, + TemplateBox[{",", "\",\"", + InterpretationBox[ + SubscriptBox["S", + TemplateBox[{",", "\",\"", "3"}, "RowWithSeparators"]], + $CellContext`S[3, Q3`$]]}, "RowWithSeparators"]]}, + "RowWithSeparators"]}, "Ket"], Ket[<|$CellContext`S[1, Q3`$] -> 1, $CellContext`S[3, Q3`$] -> 1|>]], "+", @@ -2426,30 +2511,31 @@ Cell[BoxData[ SqrtBox["2"], " ", InterpretationBox[ TemplateBox[{ - TemplateBox[{ + TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", SubscriptBox["1", - InterpretationBox[ - SubscriptBox[ - TemplateBox[{"S"}, "RowDefault"], - 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0},ExpressionUUID-> - "5ec0ecd8-aa74-4f40-a43e-5bc45068a0b1"], + "deb18a3b-9492-424c-94d2-a5b8042828ee"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -89,8 +89,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "f0e75bef-c7c4-46aa-ad76-26e6eaeb1141"] - }],ExpressionUUID->"b4297aa8-1b2c-4c43-ad1d-c39bc3f1cff7"], + "74b3bc98-42ea-4f33-a296-6ea1e4a6b4b9"] + }],ExpressionUUID->"d7c0d79e-eeb2-4b63-b1cf-c7ae018fd159"], StripOnInput->False],{ "\"Quantum Information Systems\"" :> Documentation`HelpLookup[ @@ -105,7 +105,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"MoreAbout"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "327fd334-ce25-472d-a931-c6b1d93434bc"], + "b73c94c3-e6ff-43dc-a6f2-3a2569e1f671"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -124,8 +124,8 @@ 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Closed]]]] +}, Dynamic[CurrentValue[EvaluationNotebook[], {TaggingRules, "Openers", "ExampleSection", "0"}, Closed]]]] }, Dynamic[CurrentValue[EvaluationNotebook[], {TaggingRules, "Openers", "PrimaryExamplesSection"}, Closed]]]], -Cell[118212, 2171, 2876, 78, 70, "SeeAlsoSection",ExpressionUUID->"ac37b7c9-33c6-4597-94a0-4271f4625c1c"], -Cell[121091, 2251, 2080, 53, 70, "TechNotesSection",ExpressionUUID->"e9ff7353-0670-473c-b0be-716054974ad8"], -Cell[123174, 2306, 742, 19, 70, "MoreAboutSection",ExpressionUUID->"de2a1a1a-ed93-4bec-ada8-02405a65758e"], -Cell[123919, 2327, 3865, 104, 70, "RelatedLinksSection",ExpressionUUID->"97b82f01-4fd7-48cd-b8b7-9cf72481436a"], -Cell[127787, 2433, 78, 0, 70, "FooterCell",ExpressionUUID->"50508bf1-7162-4757-999d-bc00628e572d"] +Cell[121378, 2202, 3900, 106, 70, "SeeAlsoSection",ExpressionUUID->"49be060a-a37a-4c92-96ca-d7e5b9d7979c"], +Cell[125281, 2310, 2080, 53, 70, "TechNotesSection",ExpressionUUID->"9306bf27-1449-42e2-840c-d4281e463045"], +Cell[127364, 2365, 742, 19, 70, "MoreAboutSection",ExpressionUUID->"86516cb7-9bd1-4476-ba49-6a7c3cb7cb73"], +Cell[128109, 2386, 3865, 104, 70, "RelatedLinksSection",ExpressionUUID->"d56b08da-424f-49e8-a7fa-24bc995b69c8"], +Cell[131977, 2492, 78, 0, 70, "FooterCell",ExpressionUUID->"85a5af3e-a5ec-4f6d-aaa4-73a37348466a"] } ] *) diff --git a/Q3/Documentation/English/ReferencePages/Symbols/Pfaffian.nb b/Q3/Documentation/English/ReferencePages/Symbols/Pfaffian.nb index 9a7a50d4c..b2502ade7 100644 --- a/Q3/Documentation/English/ReferencePages/Symbols/Pfaffian.nb +++ b/Q3/Documentation/English/ReferencePages/Symbols/Pfaffian.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 110566, 2850] -NotebookOptionsPosition[ 93275, 2496] -NotebookOutlinePosition[ 95623, 2556] -CellTagsIndexPosition[ 95537, 2551] +NotebookDataLength[ 140343, 3721] +NotebookOptionsPosition[ 114610, 3189] +NotebookOutlinePosition[ 116979, 3249] +CellTagsIndexPosition[ 116892, 3244] WindowTitle->Pfaffian WindowFrame->Normal*) @@ -27,11 +27,11 @@ Cell[BoxData[GridBox[{ TemplateBox[{12}, "Spacer1"], Cell["Q3 SYMBOL", "PacletNameCell", TextAlignment->Center,ExpressionUUID-> - "b1e03dc7-91f8-493d-a892-b8a114499d4f"], + "134647e8-470f-4b76-80a6-ac0ab56f3d6b"], TemplateBox[{8}, "Spacer1"]}]], TextAlignment->Center,ExpressionUUID-> - "50cd51e8-c14b-444a-8497-10ea76453de4"], + "351a1386-d824-4a42-bc01-b34f5479781c"], Background->RGBColor[0.490196, 0.576471, 0.690196], ItemSize->Full], ""} }, @@ -54,8 +54,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "9e833162-e8d9-4461-a2f8-12ccc5b25fc3"] - }],ExpressionUUID->"44996b49-4b18-4653-816b-32df34764f7c"], + "f981b08a-eb57-4769-b7ba-05a51fed46fe"] + }],ExpressionUUID->"e621accc-93fe-4c1c-9111-d2a2217b7f59"], StripOnInput->False],{ "\"Det\"" :> Documentation`HelpLookup["paclet:ref/Det"], Delimiter, StyleBox[ @@ -70,7 +70,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"SeeAlso"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "ba8051ff-80f7-4e32-8642-0056ab2c872e"], + "d0e2eb5c-0409-4d23-8fa2-c460c103cade"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -89,8 +89,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "0c9074ea-32ac-4e01-b87b-f8c6c22c273d"] - }],ExpressionUUID->"b1d4803b-ab42-4968-89c8-9dc0b9d7e17e"], + "76d98d73-5f15-41a6-85f2-4e88ce9b4cd7"] + }],ExpressionUUID->"e9eef8f6-b7c0-48ea-a652-dbff63ae43c1"], StripOnInput->False],{ "\"Quantum Information Systems\"" :> Documentation`HelpLookup[ @@ -106,7 +106,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"MoreAbout"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "778d772e-6e72-401b-b2f7-b64656a18c42"], + "91fdd873-5a7c-41f9-8959-386028271772"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -125,8 +125,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "15b51658-f315-4f05-96e6-0cfe17952330"] - }],ExpressionUUID->"e0bb87df-d8cd-45e0-8ee9-030c5a623c5b"], + "f561fc23-6230-4b82-bab3-dce5fa825f14"] + }],ExpressionUUID->"51ab0ba1-de94-4a41-8261-5927a184238e"], StripOnInput->False],{ "\"Quantum Information Systems with Q3\"" :> Documentation`HelpLookup[ @@ -141,7 +141,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"Tutorials"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "df7468ea-5128-4263-84cc-b9ca8b02bc0f"], + "1ec38e25-731c-4dd6-93d8-9e83bf74f018"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -160,8 +160,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "a75a4a00-7702-4646-8135-92374ce142e7"] - }],ExpressionUUID->"2b9ac837-2c9d-48c6-bf8b-73fc9b028a39"], + "b6aa7a88-4173-44f0-8362-db94cd33476b"] + }],ExpressionUUID->"6a0b4f66-0499-495c-b81f-a6bab315ee1b"], StripOnInput->False],{ "\"Q3/ref/Pfaffian\"" :> None, "\"Copy Wolfram Documentation Center URL\"" :> @@ -198,14 +198,14 @@ html"], StandardForm]], "Input", TextClipboardType -> "PlainText"]}, Visible -> MenuStyle->"URLMenu"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "eda3d188-32e7-4078-b33d-956f7e4f46b9"] + "b2a4bd74-ef62-4066-909b-86d932772bb6"] }], "AnchorBar", CacheGraphics->False,ExpressionUUID-> - "1f949aa8-312b-4468-b5ce-b069d814fd32"]} + "9c40eef9-af5c-4841-b965-38fd8e607afc"]} }]], "AnchorBarGrid", - CellID->1,ExpressionUUID->"88665f15-b9bb-4a37-899c-b00893133f8d"], + CellID->1,ExpressionUUID->"0590f9c8-92f8-45d9-97a7-209a45215fa0"], -Cell["Q3`", "ContextNameCell",ExpressionUUID->"47692202-8188-4e94-a653-698c7a0aef0a"], +Cell["Q3`", "ContextNameCell",ExpressionUUID->"7a2636d8-9c80-4f9b-b68d-a47ec6c318ed"], Cell[CellGroupData[{ @@ -213,14 +213,14 @@ Cell[BoxData[GridBox[{ {Cell[TextData[{ Cell[ "Pfaffian", "ObjectName",ExpressionUUID-> - "47793308-d55c-453b-92fb-a4f4d21d0b4d"], + "5122b383-ae39-4b6a-8d87-baae00ed3484"], Cell[BoxData[ TemplateBox[{8}, - "Spacer1"]],ExpressionUUID->"41d3f024-3387-428b-9b8f-61c7d0fb88ce"], + "Spacer1"]],ExpressionUUID->"223958a8-aec2-4897-a199-2dad261d3e43"], Cell[BoxData[ ""], "ObjectNameTranslation",ExpressionUUID-> - "d12f9ee2-22b2-4005-88fc-9c96230b6045"] - }],ExpressionUUID->"7796af62-5a5b-45c9-b980-36c4b37d2113"], Cell[BoxData[ + "21188ceb-c67f-4d3a-bc3d-1d7f1a896aba"] + }],ExpressionUUID->"1feba917-39b6-443a-baa7-c1d47e4ab8dd"], Cell[BoxData[ TooltipBox[ StyleBox[ TagBox[ @@ -245,9 +245,9 @@ Cell[BoxData[GridBox[{ "New in 14", TooltipDelay->0.3]], Magnification->1,ExpressionUUID-> - "0df84af5-8bd4-4eef-927c-1480c0f36c4f"]} + "4d1b44f8-ee30-43be-9359-b9acc15e1c8e"]} }]], "ObjectNameGrid", - CacheGraphics->False,ExpressionUUID->"1e06df4d-26db-44a5-9775-84209bc525d5"], + CacheGraphics->False,ExpressionUUID->"f8af762a-a89d-4790-837e-c0f249fadbd8"], Cell[BoxData[GridBox[{ {"", Cell[TextData[{ @@ -269,7 +269,7 @@ anti-symmetric) matrix ", FontFamily->"Source Sans Pro",ExpressionUUID-> "611115ae-6d4c-40df-a453-55acf7a9e542"], "." - }],ExpressionUUID->"a4d933c1-9f94-47f0-b505-f1c2b610b62e"]} + }],ExpressionUUID->"dd77daae-3fce-4614-a60d-3e88203ed781"]} }]], "Usage", CellID->1422330313,ExpressionUUID->"dbd3aa71-c713-419d-aa14-1b00ac751316"] }, Open ]], @@ -302,12 +302,12 @@ Cell[TextData[Cell[BoxData[ PlotRange -> {{-3, 4}, {-1, 1}}, ImageSize -> 20], Magnification -> 0.68 Inherited], Rational[1, 2] Pi, {-1.65, -1}]]], ImageSizeCache->{14., {0.45000000000000107`, 13.549999999999999`}}]], - ExpressionUUID->"0b3571fc-3307-4a49-ab4d-6909d0fe0cf6"], + 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CellID->2110208993,ExpressionUUID->"adbd6825-0323-417c-a99b-c931bfef015b"], Cell["\<\ The term Pfaffian was named after Johann Friedrich Pfaff by Arthur Cayley in \ @@ -451,35 +451,82 @@ Cell[TextData[{ CellID->1638344201,ExpressionUUID->"88653d8a-6f93-4c06-8ab8-dc2dd039d953"], Cell[TextData[{ + "A ", Cell[BoxData[ TemplateBox[{ Cell[ - TextData["Pfaffian"]], "paclet:Q3/ref/Pfaffian", "Q3 Package Symbol"}, - "PackageLink", - BaseStyle->"InlineFormula"]], "InlineFormula", + TextData["Method"]], "paclet:ref/Method"}, + "RefLink", + BaseStyle->{"InlineFormula"}]], "InlineFormula", FontFamily->"Source Sans Pro",ExpressionUUID-> - "a41036d3-1a9e-4f8a-b88b-f5777198c784"], - " accept the following option." + "4a2a7763-cc3f-4010-b808-671a51d9fd2a"], + " option can be given. Default setting ", + Cell[BoxData[ + RowBox[{ + TemplateBox[{ + Cell[ + TextData["Method"]], "paclet:ref/Method"}, + "RefLink", + BaseStyle->{"InlineFormula"}], "\[Rule]", + TemplateBox[{ + Cell[ + TextData["Automatic"]], "paclet:ref/Automatic"}, + "RefLink", + BaseStyle->{"InlineFormula"}]}]], "InlineFormula", + FontFamily->"Source Sans Pro",ExpressionUUID-> + "3989a6f1-bdd2-4b30-9b68-008092b591dd"], + " uses the ", + StyleBox["Hessenberg decomposition", + FontSlant->"Italic"], + " (\"Hessenberg\") for real matrices (note that it converts any exact number \ +to a finite-precision number, and hence the returned value is always an \ +inexact number) and the ", + StyleBox["Householder tridiagonalization", + FontSlant->"Italic"], + " for complex matrices. Other settings include: " }], "Notes", - CellID->1768451127,ExpressionUUID->"d544b462-9fcf-4441-ada3-43851449fcc4"], + CellID->1347491459,ExpressionUUID->"51cde008-3b79-47a0-80d8-28cf98908082"], Cell[BoxData[GridBox[{ {Cell[" ", "TableRowIcon",ExpressionUUID-> - "bc176b44-da61-49e2-bf05-bd40f51f0c06"], - TemplateBox[{ - Cell[ - TextData["Method"]], "paclet:ref/Method"}, - "RefLink", - BaseStyle->{"3ColumnTableMod"}], "\"\\"", Cell["\<\ -Using the Parlett-Reid tridiagonalization. \"Householder\" uses the \ -Householder tridiagonalization, \"Hessenberg\" uses the Hessenberg \ -decomposition, and \"Heuristic\" uses the heuristic recursive method.\ -\>", "TableText",ExpressionUUID->"884cb493-9ae7-4c62-bd88-a9e0aac0eacf"]} - }]], "3ColumnTableMod", + "bbc9a570-c546-4719-a865-d45cfa94cbd7"], Cell[ + "\"Parlett-Reid\"", "TableText",ExpressionUUID-> + "11267bbd-d7f1-422f-a179-1920c405d74f"], Cell[ + "Uses the Parlett-Reid tridiagonalization.", "TableText",ExpressionUUID-> + "94478be4-b597-4f7a-8778-f17dd9c33313"]}, + {Cell[" ", "TableRowIcon",ExpressionUUID-> + "81747c17-ba0a-4295-a232-502abcf7feb7"], Cell[ + "\"Householder\"", "TableText",ExpressionUUID-> + "8297c162-5153-42e6-9ddb-45080ff017f4"], Cell["\<\ +Uses the Householder tridiagonalization, and is the default choice for \ +complex matrices.\ +\>", "TableText",ExpressionUUID->"c7767a13-09e3-4855-90f8-c2095dcefe5f"]}, + {Cell[" ", "TableRowIcon",ExpressionUUID-> + "305a26b8-9653-41f0-ba41-73a22f44b2d8"], Cell[ + "\"Hessenberg\"", "TableText",ExpressionUUID-> + "723992e6-5e7e-4745-bfc6-f91f0bb31191"], Cell[TextData[{ + "Uses the Hessenberg decomposition, is usually the fastest for ", + StyleBox["real", + FontSlant->"Italic"], + " matrices, and hence is the default choice for ", + StyleBox["real", + FontSlant->"Italic"], + " matrices. However, it works only for finite-preision matrices. For \ +infinite-precision real matrices, use \"Householder\"." + }], "TableText",ExpressionUUID->"d119cba5-6f3a-4d5e-94fc-1e617623dd58"]}, + {Cell[" ", "TableRowIcon",ExpressionUUID-> + "f74dbdee-1524-41db-9aee-793a4feb688a"], Cell[ + "\"Heuristic\"", "TableText",ExpressionUUID-> + "5baf5673-96a0-4593-bc2a-dbb5009858cc"], Cell["\<\ +Uses the heuristic recursive method. 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a/Q3/Documentation/English/ReferencePages/Symbols/SkewTridiagonalize.nb b/Q3/Documentation/English/ReferencePages/Symbols/SkewTridiagonalize.nb index 61b54eaf0..a5834aab8 100644 --- a/Q3/Documentation/English/ReferencePages/Symbols/SkewTridiagonalize.nb +++ b/Q3/Documentation/English/ReferencePages/Symbols/SkewTridiagonalize.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 89338, 2276] -NotebookOptionsPosition[ 77161, 2031] -NotebookOutlinePosition[ 79579, 2090] -CellTagsIndexPosition[ 79493, 2085] +NotebookDataLength[ 93835, 2393] +NotebookOptionsPosition[ 81198, 2140] +NotebookOutlinePosition[ 83618, 2199] +CellTagsIndexPosition[ 83532, 2194] WindowTitle->SkewTridiagonalize WindowFrame->Normal*) @@ -27,11 +27,11 @@ Cell[BoxData[GridBox[{ TemplateBox[{12}, "Spacer1"], Cell["Q3 SYMBOL", "PacletNameCell", TextAlignment->Center,ExpressionUUID-> - "550c9899-f883-4427-868f-526cd7283878"], + "2d68da29-c65e-4905-8b66-067ac26bb67e"], TemplateBox[{8}, "Spacer1"]}]], TextAlignment->Center,ExpressionUUID-> - "9aebb3dd-bbb3-40ea-bd6a-6e5d1c94d761"], + "ab4c82df-5d67-4e2a-b732-8ade4de9d240"], Background->RGBColor[0.490196, 0.576471, 0.690196], ItemSize->Full], ""} }, @@ -54,8 +54,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "41145d82-14a6-4458-b3b8-aabb35db1f11"] - }],ExpressionUUID->"465147c3-00f9-4f29-928d-6fe21c43bd03"], + "41cdd3bd-14fd-41ea-814c-4dcdbb01bb94"] + }],ExpressionUUID->"68431897-869e-4711-82b0-642b92c6a1bd"], StripOnInput->False],{ "\"HessenbergDecomposition\"" :> Documentation`HelpLookup["paclet:ref/HessenbergDecomposition"], @@ -71,7 +71,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"SeeAlso"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "3a3c5620-69a1-47f9-b0b8-53a0902c49d2"], + "c2cdc397-ecfa-4516-99de-f5fc4e6a8973"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -90,8 +90,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "b6a60ce6-cbb0-488d-95e2-8f04b990ba86"] - }],ExpressionUUID->"5fa73b2f-4061-4031-aa39-ac93d01b5b21"], + "d6566486-b466-4c46-b391-d48578aeda14"] + }],ExpressionUUID->"2e16c3cd-cb39-47d7-81b4-2c07bd777bec"], StripOnInput->False],{ "\"Quantum Information Systems\"" :> Documentation`HelpLookup[ @@ -107,7 +107,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"MoreAbout"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "e440790b-063b-49fa-96fc-a1465ad9cb9e"], + "6c903980-b837-491d-a9b4-e1495e9da86f"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -126,8 +126,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - 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Default setting ", + Cell[BoxData[ + RowBox[{ + TemplateBox[{ + Cell[ + TextData["Method"]], "paclet:ref/Method"}, + "RefLink", + BaseStyle->{"InlineFormula"}], "\[Rule]", + TemplateBox[{ + Cell[ + TextData["Automatic"]], "paclet:ref/Automatic"}, + "RefLink", + BaseStyle->{"InlineFormula"}]}]], "InlineFormula", + FontFamily->"Source Sans Pro",ExpressionUUID-> + "01083d96-dcf2-491f-b2de-f2c0e0210d5a"], + " uses the ", + StyleBox["Hessenberg decomposition", + FontSlant->"Italic"], + " (\"Hessenberg\") for real matrices (note that it converts any exact number \ +to a finite-precision number, and hence the returned value is always an \ +inexact number) and the ", + StyleBox["Householder tridiagonalization", + FontSlant->"Italic"], + " for complex matrices. Other settings include: " +}], "Notes", + CellID->1347491459,ExpressionUUID->"003869a8-71ba-4699-9f57-56c2f11ba284"], Cell[BoxData[GridBox[{ {Cell[" ", "TableRowIcon",ExpressionUUID-> - "6a85c24b-7028-4bcd-8a2d-3f111d82b129"], - TemplateBox[{ - Cell[ - TextData["Method"]], "paclet:ref/Method"}, - "RefLink", - BaseStyle->{"3ColumnTableMod"}], "\"\\"", Cell["\<\ -\"ParlettReid\" uses the Parlett-Reid tridiagonalization. Other methods \ -include \"Householder\" based on the Householder tridiagonalization and \ -\"Hessenberg\" based on the Hessenberg decomposition.\ -\>", "TableText",ExpressionUUID->"b30c3d25-8977-48cf-9365-ac35690471dd"]} - }]], "3ColumnTableMod", + "06c11c35-209a-478b-a7e1-8d953871c8fa"], Cell[ + "\"Parlett-Reid\"", "TableText",ExpressionUUID-> + "032dabf8-bdc7-4764-993d-1784b197ddb0"], Cell[ + "Uses the Parlett-Reid tridiagonalization.", "TableText",ExpressionUUID-> + "02d0e403-d4f7-41cd-8097-2229e439a4b6"]}, + {Cell[" ", "TableRowIcon",ExpressionUUID-> + "f259393e-f9f2-4529-883f-5e22e0dc7b4d"], Cell[ + "\"Householder\"", "TableText",ExpressionUUID-> + "e93b3e6d-e444-41e8-ae94-723322b9cd5b"], Cell["\<\ +Uses the Householder tridiagonalization, and is the default choice for \ +complex matrices.\ +\>", "TableText",ExpressionUUID->"95a65c8f-5bc4-4623-b67b-cacecfa06e72"]}, + {Cell[" ", "TableRowIcon",ExpressionUUID-> + "97b1a5cc-8d1e-4c82-86a5-2f019399de5a"], Cell[ + "\"Hessenberg\"", "TableText",ExpressionUUID-> + "ebe0f332-8b7d-4908-bab4-1a63155aee3b"], Cell[TextData[{ + "Uses the Hessenberg decomposition, is usually the fastest for ", + StyleBox["real", + FontSlant->"Italic"], + " matrices, and hence is the default choice for ", + StyleBox["real", + FontSlant->"Italic"], + " matrices. However, it works only for finite-preision matrices. For \ +infinite-precision real matrices, use \"Householder\"." + }], "TableText",ExpressionUUID->"7a224475-a5cd-4fc5-8f5d-730d9d9b6554"]}, + {Cell[" ", "TableRowIcon",ExpressionUUID-> + "18e603c9-0daa-4b15-863b-301968b565a4"], Cell[ + "\"Heuristic\"", "TableText",ExpressionUUID-> + "1e99bde9-2ed3-46d1-a4d0-5729e66325e1"], Cell["\<\ +Uses the heuristic recursive method. 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"RowWithSeparators"]], - $CellContext`c[4]]}], "-", + $CellContext`c[4]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.11771855377041823`", "\[VeryThinSpace]", "-", - RowBox[{"0.11667010594161487`", " ", "\[ImaginaryI]"}]}], ")"}], " ", - + RowBox[{"0.1639656612562539`", "\[VeryThinSpace]", "-", + RowBox[{"0.2106214952178339`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{"c", TemplateBox[{",", "\",\"", "1"}, "RowWithSeparators"], TemplateBox[{",", "\",\"", "\"\[Dagger]\""}, "RowWithSeparators"]}, "Subsuperscript"], Q3`Dagger[ - $CellContext`c[1]]]}], "+", + $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.16056049965561614`", "\[VeryThinSpace]", "+", - RowBox[{"0.20945129228355727`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.0604526288077517`", "\[VeryThinSpace]", "+", + RowBox[{"0.19456572511539583`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{"c", @@ -2981,8 +4131,8 @@ Cell[BoxData[ $CellContext`c[2]]]}], "+", 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"\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{"c", @@ -3166,19 +4315,19 @@ Cell[BoxData[ $CellContext`c[2]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.020739116787077506`", "\[VeryThinSpace]", "+", - RowBox[{"0.2120262479987784`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.03602669929767887`", "\[VeryThinSpace]", "+", + RowBox[{"0.0444234159130101`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{"c", TemplateBox[{",", "\",\"", "3"}, "RowWithSeparators"], TemplateBox[{",", "\",\"", "\"\[Dagger]\""}, "RowWithSeparators"]}, "Subsuperscript"], Q3`Dagger[ - $CellContext`c[3]]]}], "+", + $CellContext`c[3]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.1499839417213059`", "\[VeryThinSpace]", "+", - RowBox[{"0.23116113409360578`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.2685838046029433`", "\[VeryThinSpace]", "-", + RowBox[{"0.05326271425192662`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{"c", @@ -3197,8 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RowBox[{"0.008117524933198617`", "\[VeryThinSpace]", "-", + RowBox[{"0.2404635076704802`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ SubscriptBox["c", TemplateBox[{",", "\",\"", "3"}, @@ -3227,17 +4375,18 @@ Cell[BoxData[ $CellContext`c[3]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.6786346138658209`", "\[VeryThinSpace]", "-", - RowBox[{"0.1737639838695046`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.08076370866206278`", "\[VeryThinSpace]", "+", + RowBox[{"0.360321020962974`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ SubscriptBox["c", TemplateBox[{",", "\",\"", "4"}, "RowWithSeparators"]], - $CellContext`c[4]]}], "+", + $CellContext`c[4]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.06039353113303273`", "\[VeryThinSpace]", "-", - RowBox[{"0.0809473511462467`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.32899932628044104`", "\[VeryThinSpace]", "-", + RowBox[{"0.29056665039401175`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + InterpretationBox[ 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"\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.05432023231621437`", "\[VeryThinSpace]", "-", + RowBox[{"0.0005108770044727137`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -3912,8 +5068,8 @@ Cell[BoxData[ "Ket"], Ket[<|$CellContext`c[1] -> 1, $CellContext`c[2] -> 1, $CellContext`c[3] -> 1, $CellContext`c[4] -> 1|>]]}]}]], "Output", - CellLabel->"Out[11]=", - CellID->1013280873,ExpressionUUID->"f556ce45-ef45-462f-9d06-d06e0cdafe3b"] + CellLabel->"Out[17]=", + CellID->1220789493,ExpressionUUID->"0c085668-2102-462f-b0f4-b03c7ac46ca5"] }, Open ]], Cell["\<\ @@ -3927,23 +5083,25 @@ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"EchoTiming", "[", RowBox[{"alt", "=", - RowBox[{"msr", "/.", "T2"}]}], "]"}]], "Input", + RowBox[{ + RowBox[{"Multiply", "@@", + RowBox[{"First", "[", "obs", "]"}]}], "/.", "T2"}]}], "]"}]], "Input", CellProlog:>Needs["Q3`"], - CellLabel->"In[12]:=", + CellLabel->"In[18]:=", CellID->1302208119,ExpressionUUID->"15601029-728a-40b2-9b82-b75709dc465a"], -Cell[BoxData["2.587838`"], "EchoTiming", - CellID->493757807,ExpressionUUID->"4f783906-c62c-44b8-a995-cde188936094"], +Cell[BoxData["2.569262`"], "EchoTiming", + CellID->1885490681,ExpressionUUID->"a9903945-073a-44ce-b05d-2d1a5af46803"], Cell[BoxData[ RowBox[{ RowBox[{"(", - RowBox[{"0.2488115368607372`", "\[VeryThinSpace]", "+", - RowBox[{"0.07666646961098034`", " ", "\[ImaginaryI]"}]}], ")"}], "-", + RowBox[{"0.009461723620606627`", "\[VeryThinSpace]", "-", + RowBox[{"0.004629775919699861`", " ", "\[ImaginaryI]"}]}], ")"}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.05050868411017906`", "\[VeryThinSpace]", "+", - RowBox[{"0.12609112273407772`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.002555288093396668`", "\[VeryThinSpace]", "-", + RowBox[{"0.01752788740145164`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -3961,8 +5119,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.13663679561595543`", "\[VeryThinSpace]", "+", - RowBox[{"0.18438944291157863`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.0014201666533639606`", "\[VeryThinSpace]", "-", + RowBox[{"0.0036213548980205956`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -3980,8 +5138,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.08981947810197946`", "\[VeryThinSpace]", "-", - RowBox[{"0.13996581051362522`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.015480527329548969`", "\[VeryThinSpace]", "+", + RowBox[{"0.009180653791880274`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -3999,8 +5157,8 @@ Cell[BoxData[ $CellContext`c[2]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.0959573004417222`", "\[VeryThinSpace]", "-", - RowBox[{"0.14944310600120497`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.011499933058205944`", "\[VeryThinSpace]", "-", + RowBox[{"0.004127916766199509`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4015,11 +5173,11 @@ Cell[BoxData[ "RowDefault"], Q3`Multiply[ $CellContext`c[4], - $CellContext`c[1]]]}], "-", + $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.08217237854840631`", "\[VeryThinSpace]", "+", - RowBox[{"0.0024536889644609216`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.004633105595028525`", "\[VeryThinSpace]", "+", + RowBox[{"0.007392436622242288`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4037,8 +5195,8 @@ Cell[BoxData[ $CellContext`c[2]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.02618206130269196`", "\[VeryThinSpace]", "+", - RowBox[{"0.02713947590493539`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.001074681036473013`", "\[VeryThinSpace]", "+", + RowBox[{"0.0023587462511685045`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4056,8 +5214,8 @@ Cell[BoxData[ $CellContext`c[3]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.09623439614612261`", "\[VeryThinSpace]", "-", - RowBox[{"0.04707710796548616`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.10688311758553604`", "\[VeryThinSpace]", "-", + RowBox[{"0.03402522186581991`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4076,11 +5234,11 @@ Cell[BoxData[ Q3`Multiply[ Q3`Dagger[ $CellContext`c[1]], - $CellContext`c[1]]]}], "+", + $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.07694552700263824`", "\[VeryThinSpace]", "+", - RowBox[{"0.1431016802221084`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.12146079386721861`", "\[VeryThinSpace]", "+", + RowBox[{"0.03048307387126327`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4099,11 +5257,11 @@ Cell[BoxData[ Q3`Multiply[ Q3`Dagger[ $CellContext`c[1]], - $CellContext`c[2]]]}], "-", + $CellContext`c[2]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.04617248640896909`", "\[VeryThinSpace]", "+", - RowBox[{"0.1080390303964251`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.06458140874044443`", "\[VeryThinSpace]", "-", + RowBox[{"0.050932185784895195`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4122,11 +5280,11 @@ Cell[BoxData[ Q3`Multiply[ Q3`Dagger[ $CellContext`c[1]], - $CellContext`c[3]]]}], "-", + $CellContext`c[3]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.03565681611321904`", "\[VeryThinSpace]", "-", - RowBox[{"0.030568165398268766`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.007956959095574603`", "\[VeryThinSpace]", "-", + RowBox[{"0.023643385037842093`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4148,8 +5306,8 @@ Cell[BoxData[ $CellContext`c[4]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.1458670570067097`", "\[VeryThinSpace]", "-", - RowBox[{"0.11228323783019502`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.05298893022726019`", "\[VeryThinSpace]", "-", + RowBox[{"0.09120595249629146`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4175,8 +5333,8 @@ Cell[BoxData[ $CellContext`c[2]]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.19238471478982064`", "\[VeryThinSpace]", "-", - RowBox[{"0.21583490478966838`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.030849004590021184`", "\[VeryThinSpace]", "-", + RowBox[{"0.1338080106606379`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4202,8 +5360,8 @@ Cell[BoxData[ $CellContext`c[3]]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.07398735436565188`", "\[VeryThinSpace]", "+", - RowBox[{"0.09339378349972602`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.0295190460265346`", "\[VeryThinSpace]", "+", + RowBox[{"0.02850989749159999`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4226,11 +5384,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[1]], Q3`Dagger[ - $CellContext`c[4]]]]}], "+", + $CellContext`c[4]]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.21734203599082638`", "\[VeryThinSpace]", "-", - RowBox[{"0.10236718730787216`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.043520708272749564`", "\[VeryThinSpace]", "-", + RowBox[{"0.06919880373934632`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4252,8 +5410,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.050308074913753664`", "\[VeryThinSpace]", "+", - RowBox[{"0.08543721454935578`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.09913271511369716`", "\[VeryThinSpace]", "-", + RowBox[{"0.05022938918189562`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4272,11 +5430,11 @@ Cell[BoxData[ Q3`Multiply[ Q3`Dagger[ $CellContext`c[2]], - $CellContext`c[2]]]}], "-", + $CellContext`c[2]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.14353632959779142`", "\[VeryThinSpace]", "+", - RowBox[{"0.006509338993111628`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.012263031026474074`", "\[VeryThinSpace]", "-", + RowBox[{"0.06492499752652964`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4295,11 +5453,11 @@ Cell[BoxData[ Q3`Multiply[ Q3`Dagger[ $CellContext`c[2]], - $CellContext`c[3]]]}], "+", + $CellContext`c[3]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.01473886379680852`", "\[VeryThinSpace]", "+", - RowBox[{"0.055512873702631235`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.007650543823429931`", "\[VeryThinSpace]", "+", + RowBox[{"0.01852242087296077`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4318,11 +5476,11 @@ Cell[BoxData[ Q3`Multiply[ Q3`Dagger[ $CellContext`c[2]], - $CellContext`c[4]]]}], "+", + $CellContext`c[4]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.08721955689245112`", "\[VeryThinSpace]", "-", - RowBox[{"0.25436541255492356`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.016424386808997285`", "\[VeryThinSpace]", "-", + RowBox[{"0.07905841604425858`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4345,11 +5503,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[2]], Q3`Dagger[ - $CellContext`c[3]]]]}], "-", + $CellContext`c[3]]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.078649799701171`", "\[VeryThinSpace]", "-", - RowBox[{"0.07200676221284251`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.07501300670546257`", "\[VeryThinSpace]", "+", + RowBox[{"0.08341365779548554`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4375,8 +5533,8 @@ Cell[BoxData[ $CellContext`c[4]]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.15606890366233928`", "\[VeryThinSpace]", "-", - RowBox[{"0.037993594420661955`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.025457473923794854`", "\[VeryThinSpace]", "+", + RowBox[{"0.06724279641079153`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4395,11 +5553,11 @@ Cell[BoxData[ Q3`Multiply[ Q3`Dagger[ $CellContext`c[3]], - $CellContext`c[1]]]}], "+", + $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.13121080201494814`", "\[VeryThinSpace]", "+", - RowBox[{"0.020801223081051222`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.01558188607033531`", "\[VeryThinSpace]", "-", + RowBox[{"0.08709944147829345`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4421,8 +5579,8 @@ Cell[BoxData[ $CellContext`c[2]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.3419703790287689`", "\[VeryThinSpace]", "+", - RowBox[{"0.10012651005463899`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.04850366650946464`", "\[VeryThinSpace]", "+", + RowBox[{"0.03841708349688351`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4441,11 +5599,11 @@ Cell[BoxData[ Q3`Multiply[ Q3`Dagger[ $CellContext`c[3]], - $CellContext`c[3]]]}], "+", + $CellContext`c[3]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.002231284164168644`", "\[VeryThinSpace]", "+", - RowBox[{"0.038337090064438584`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.017053021409822344`", "\[VeryThinSpace]", "+", + RowBox[{"0.004459271971292589`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4464,11 +5622,11 @@ Cell[BoxData[ Q3`Multiply[ Q3`Dagger[ $CellContext`c[3]], - $CellContext`c[4]]]}], "-", + $CellContext`c[4]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.07885000318299294`", "\[VeryThinSpace]", "-", - RowBox[{"0.14891997486212272`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.0877830942539381`", "\[VeryThinSpace]", "+", + RowBox[{"0.07528886478777239`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4491,11 +5649,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[3]], Q3`Dagger[ - $CellContext`c[4]]]]}], "-", + $CellContext`c[4]]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.03966472947985955`", "\[VeryThinSpace]", "+", - RowBox[{"0.054767652822000606`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.03276309532702279`", "\[VeryThinSpace]", "-", + RowBox[{"0.09569066751622962`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4514,11 +5672,11 @@ Cell[BoxData[ Q3`Multiply[ Q3`Dagger[ $CellContext`c[4]], - $CellContext`c[1]]]}], "-", + $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.012793246199766774`", "\[VeryThinSpace]", "+", - RowBox[{"0.054445464443274125`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.09504088365539828`", "\[VeryThinSpace]", "-", + RowBox[{"0.0803742905973556`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4540,8 +5698,8 @@ Cell[BoxData[ $CellContext`c[2]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.005696164623878496`", "\[VeryThinSpace]", "+", - RowBox[{"0.040040295694878505`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.003666070576981359`", "\[VeryThinSpace]", "+", + RowBox[{"0.08166891916923553`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4560,11 +5718,11 @@ Cell[BoxData[ Q3`Multiply[ Q3`Dagger[ $CellContext`c[4]], - $CellContext`c[3]]]}], "-", + $CellContext`c[3]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.2342271563199968`", "\[VeryThinSpace]", "+", - RowBox[{"0.08364235935077999`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.005010621645680696`", "\[VeryThinSpace]", "+", + RowBox[{"0.024763528561322225`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4583,11 +5741,11 @@ Cell[BoxData[ Q3`Multiply[ Q3`Dagger[ $CellContext`c[4]], - $CellContext`c[4]]]}], "-", + $CellContext`c[4]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.01602278489972652`", "\[VeryThinSpace]", "+", - RowBox[{"0.05952438690344332`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.009273260021583902`", "\[VeryThinSpace]", "+", + RowBox[{"0.010959623793443424`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4612,11 +5770,11 @@ Cell[BoxData[ $CellContext`c[4], $CellContext`c[3], $CellContext`c[2], - $CellContext`c[1]]]}], "+", + $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.0038897390217550783`", "\[VeryThinSpace]", "-", - RowBox[{"0.020345897466225667`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.09870356222399371`", 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"\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.09234369302310148`", "\[VeryThinSpace]", "-", + RowBox[{"0.015518989408014532`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4711,11 +5869,11 @@ Cell[BoxData[ $CellContext`c[1]], $CellContext`c[4], $CellContext`c[3], - $CellContext`c[1]]]}], "-", + $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.01068512975097349`", "\[VeryThinSpace]", "-", - RowBox[{"0.02170069880910254`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.0020117868028456283`", "\[VeryThinSpace]", "-", + RowBox[{"0.08923769527545722`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4744,11 +5902,11 @@ Cell[BoxData[ $CellContext`c[1]], $CellContext`c[4], $CellContext`c[3], - $CellContext`c[2]]]}], "-", + $CellContext`c[2]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.008869539551777838`", "\[VeryThinSpace]", "+", - RowBox[{"0.015957518087033037`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.12025784168659535`", "\[VeryThinSpace]", "-", + RowBox[{"0.239751696711922`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4784,8 +5942,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.01868555949593804`", "\[VeryThinSpace]", "-", - RowBox[{"0.010056827101822874`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.00392679650546774`", "\[VeryThinSpace]", "-", + RowBox[{"0.1029745418405424`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4821,8 +5979,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.002079014797863061`", "\[VeryThinSpace]", "-", - RowBox[{"0.0009636799423418446`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.11538231563283295`", "\[VeryThinSpace]", "+", + RowBox[{"0.04118221810522309`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4855,11 +6013,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[2]], $CellContext`c[3], - $CellContext`c[2]]]}], "+", + $CellContext`c[2]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.03231736703127898`", "\[VeryThinSpace]", "+", - RowBox[{"0.060902899151191264`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.06509809656726087`", "\[VeryThinSpace]", "-", + RowBox[{"0.11689628757801354`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -4895,8 +6053,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.0006794883231384212`", "\[VeryThinSpace]", "+", - RowBox[{"0.01585354285284055`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.07175830065130048`", 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TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5007,11 +6165,11 @@ Cell[BoxData[ $CellContext`c[2]], Q3`Dagger[ $CellContext`c[3]], - $CellContext`c[1]]]}], "-", + $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.024893041800261466`", "\[VeryThinSpace]", "-", - RowBox[{"0.0037676345920158`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.09448506767310426`", "\[VeryThinSpace]", "+", + RowBox[{"0.011987623768999034`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5051,8 +6209,8 @@ Cell[BoxData[ $CellContext`c[2]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.011530513669368444`", "\[VeryThinSpace]", "+", - RowBox[{"0.02963300725771544`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.06605422705752384`", "\[VeryThinSpace]", "+", + RowBox[{"0.022844018716376504`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5092,8 +6250,8 @@ Cell[BoxData[ $CellContext`c[3]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.07023134040104773`", "\[VeryThinSpace]", "+", - RowBox[{"0.034197361519359516`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.022228485069324122`", "\[VeryThinSpace]", "+", + RowBox[{"0.026799174518203704`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5130,11 +6288,11 @@ Cell[BoxData[ $CellContext`c[2]], Q3`Dagger[ $CellContext`c[3]], - $CellContext`c[4]]]}], "+", + $CellContext`c[4]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.08144696228395534`", "\[VeryThinSpace]", "-", - RowBox[{"0.06698238536005738`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.0006500184205057991`", "\[VeryThinSpace]", "+", + RowBox[{"0.16749448967469616`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5178,8 +6336,8 @@ Cell[BoxData[ $CellContext`c[4]]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.057387789724363855`", "\[VeryThinSpace]", "+", - RowBox[{"0.061981928004307206`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.0890284752113622`", "\[VeryThinSpace]", "-", + RowBox[{"0.05088546333478779`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5219,8 +6377,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.033015244320475076`", "\[VeryThinSpace]", "-", - RowBox[{"0.057641657906597886`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.07010548004235498`", "\[VeryThinSpace]", "-", + RowBox[{"0.16150261508754513`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5257,11 +6415,11 @@ Cell[BoxData[ $CellContext`c[2]], Q3`Dagger[ $CellContext`c[4]], - $CellContext`c[2]]]}], "+", + $CellContext`c[2]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.07475713342315128`", "\[VeryThinSpace]", "+", - RowBox[{"0.032664018838417325`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.029189696697746487`", "\[VeryThinSpace]", "+", + RowBox[{"0.10516282528406273`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5301,8 +6459,8 @@ Cell[BoxData[ $CellContext`c[3]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.14935311338347107`", "\[VeryThinSpace]", "-", - RowBox[{"0.14470782167285598`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.028138850031784554`", "\[VeryThinSpace]", "-", + RowBox[{"0.1130745055338894`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5339,11 +6497,11 @@ Cell[BoxData[ $CellContext`c[2]], Q3`Dagger[ $CellContext`c[4]], - $CellContext`c[4]]]}], "+", + $CellContext`c[4]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.019458266218396216`", "\[VeryThinSpace]", "-", - RowBox[{"0.013545923988003793`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.1408632611079967`", "\[VeryThinSpace]", "+", + RowBox[{"0.25693550253204767`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5376,11 +6534,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[3]], $CellContext`c[2], - $CellContext`c[1]]]}], "-", + $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.05994334220791302`", "\[VeryThinSpace]", "-", - RowBox[{"0.029165994417091817`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.17091893444042644`", "\[VeryThinSpace]", "+", + RowBox[{"0.05314265832155632`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5413,11 +6571,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[3]], $CellContext`c[3], - $CellContext`c[1]]]}], "+", + $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.022895640031283658`", "\[VeryThinSpace]", "+", - RowBox[{"0.012272912651755112`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.0986269405680269`", "\[VeryThinSpace]", "-", + RowBox[{"0.11178407913361643`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5450,11 +6608,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[3]], $CellContext`c[3], - $CellContext`c[2]]]}], "+", + $CellContext`c[2]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.1043443986254492`", "\[VeryThinSpace]", "+", - RowBox[{"0.12761471146842468`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.03561753789460759`", "\[VeryThinSpace]", "-", + RowBox[{"0.15477780360483195`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5487,11 +6645,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[3]], $CellContext`c[4], - $CellContext`c[1]]]}], "+", + $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.04044582614633686`", "\[VeryThinSpace]", "-", - RowBox[{"0.08184690391291011`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.11288811498038157`", "\[VeryThinSpace]", "+", + RowBox[{"0.013818766013339482`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5527,8 +6685,8 @@ Cell[BoxData[ $CellContext`c[2]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.0751607426288476`", "\[VeryThinSpace]", "-", - RowBox[{"0.044467878419245495`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.024747584109926245`", "\[VeryThinSpace]", "+", + RowBox[{"0.03213853371945917`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5561,11 +6719,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[3]], $CellContext`c[4], - $CellContext`c[3]]]}], "-", + $CellContext`c[3]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.013370405504908436`", "\[VeryThinSpace]", "+", - RowBox[{"0.10774253568527845`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.1046897772947791`", "\[VeryThinSpace]", "+", + RowBox[{"0.00524366997722049`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5602,11 +6760,11 @@ Cell[BoxData[ $CellContext`c[3]], Q3`Dagger[ $CellContext`c[4]], - $CellContext`c[1]]]}], "-", + $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.08350662118714507`", "\[VeryThinSpace]", "-", - RowBox[{"0.001998545651498434`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.15242363146493082`", "\[VeryThinSpace]", "+", + RowBox[{"0.20662064220939116`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5643,11 +6801,11 @@ Cell[BoxData[ $CellContext`c[3]], Q3`Dagger[ $CellContext`c[4]], - $CellContext`c[2]]]}], "+", + $CellContext`c[2]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.13426107768546064`", "\[VeryThinSpace]", "+", - RowBox[{"0.0973501600996505`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.13907937661303532`", "\[VeryThinSpace]", "-", + RowBox[{"0.024877570985160573`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5684,11 +6842,11 @@ Cell[BoxData[ $CellContext`c[3]], Q3`Dagger[ $CellContext`c[4]], - $CellContext`c[3]]]}], "+", + $CellContext`c[3]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.18666191055086706`", "\[VeryThinSpace]", "-", - RowBox[{"0.23929257869513437`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.012572525243319796`", "\[VeryThinSpace]", "+", + RowBox[{"0.10677574612322328`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5725,11 +6883,11 @@ Cell[BoxData[ $CellContext`c[3]], Q3`Dagger[ $CellContext`c[4]], - $CellContext`c[4]]]}], "+", + $CellContext`c[4]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.02641490018570789`", "\[VeryThinSpace]", "-", - RowBox[{"0.006205540733154227`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.16263414447170987`", "\[VeryThinSpace]", "-", + RowBox[{"0.09428442656868335`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5762,11 +6920,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[4]], $CellContext`c[2], - $CellContext`c[1]]]}], "-", + $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.0019363652205291564`", "\[VeryThinSpace]", "-", - RowBox[{"0.06491972402379606`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.011483561457391527`", "\[VeryThinSpace]", "-", + RowBox[{"0.08127477688291447`", " ", 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InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5873,11 +7031,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[4]], $CellContext`c[4], - $CellContext`c[1]]]}], "-", + $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.10730585622847522`", "\[VeryThinSpace]", "+", - RowBox[{"0.16532055358506792`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.10053467454545659`", "\[VeryThinSpace]", "+", + RowBox[{"0.05726717889433066`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5910,11 +7068,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[4]], $CellContext`c[4], - $CellContext`c[2]]]}], "+", + $CellContext`c[2]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.04901594362427766`", "\[VeryThinSpace]", "+", - RowBox[{"0.1250608373962332`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.07179427279179551`", "\[VeryThinSpace]", "-", + RowBox[{"0.05799992985813318`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5947,11 +7105,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[4]], $CellContext`c[4], - $CellContext`c[3]]]}], "+", + $CellContext`c[3]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.009799605925303838`", "\[VeryThinSpace]", "-", - RowBox[{"0.003973861254521827`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.046722286105594764`", "\[VeryThinSpace]", "-", + RowBox[{"0.04289809884431299`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -5983,8 +7141,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.0254759705707471`", "\[VeryThinSpace]", "+", - RowBox[{"0.007809606498247593`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.09291137866758212`", "\[VeryThinSpace]", "-", + RowBox[{"0.06812435389517421`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6013,11 +7171,11 @@ Cell[BoxData[ $CellContext`c[2]], $CellContext`c[4], $CellContext`c[2], - $CellContext`c[1]]]}], "+", + $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.04677029679824423`", "\[VeryThinSpace]", "-", - RowBox[{"0.04376988687511922`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.0458684932638525`", "\[VeryThinSpace]", "-", + RowBox[{"0.05897506818520917`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6049,8 +7207,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.007950640650048338`", "\[VeryThinSpace]", "+", - RowBox[{"0.003844639550587378`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.04692744322553974`", "\[VeryThinSpace]", "+", + RowBox[{"0.05748794443831046`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6079,11 +7237,11 @@ Cell[BoxData[ $CellContext`c[2]], $CellContext`c[4], $CellContext`c[3], - $CellContext`c[2]]]}], "+", + $CellContext`c[2]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.0242630388992948`", "\[VeryThinSpace]", "+", - RowBox[{"0.010055386324352373`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.06868655403111407`", "\[VeryThinSpace]", "+", + RowBox[{"0.030600643384853927`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6116,11 +7274,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[3]], $CellContext`c[2], - $CellContext`c[1]]]}], "-", + $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.02255828498176362`", "\[VeryThinSpace]", "+", - RowBox[{"0.0327016777942336`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.0587785395875028`", "\[VeryThinSpace]", "-", + RowBox[{"0.0435803213698601`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6156,8 +7314,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.0018922051704797485`", "\[VeryThinSpace]", "+", - RowBox[{"0.01163210590516044`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.009827910941713904`", "\[VeryThinSpace]", "+", + RowBox[{"0.04342798280658971`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6193,8 +7351,8 @@ Cell[BoxData[ $CellContext`c[2]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.1511591307349999`", "\[VeryThinSpace]", "+", - RowBox[{"0.10390893936773396`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.02904181817397441`", "\[VeryThinSpace]", "-", + RowBox[{"0.0890438060988278`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6230,8 +7388,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.007431153959350425`", "\[VeryThinSpace]", "-", - RowBox[{"0.04641162067274981`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.04948907198033994`", "\[VeryThinSpace]", "+", + RowBox[{"0.002220824911108782`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6264,11 +7422,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[3]], $CellContext`c[4], - $CellContext`c[2]]]}], "+", + $CellContext`c[2]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.0436052673186898`", "\[VeryThinSpace]", "+", - RowBox[{"0.029365360237024865`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.026985739504758313`", "\[VeryThinSpace]", "+", + RowBox[{"0.023098331713444525`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6301,11 +7459,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[3]], $CellContext`c[4], - $CellContext`c[3]]]}], "-", + $CellContext`c[3]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.0001129316323702674`", "\[VeryThinSpace]", "-", - RowBox[{"0.12673528683443108`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.1533019900547642`", "\[VeryThinSpace]", "+", + RowBox[{"0.05304401974770509`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6345,8 +7503,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.038330017675917774`", "\[VeryThinSpace]", "-", - RowBox[{"0.05224991911279084`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.04063023552635277`", "\[VeryThinSpace]", "+", + RowBox[{"0.0775589966952406`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6386,8 +7544,8 @@ Cell[BoxData[ $CellContext`c[2]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.043219430286656514`", "\[VeryThinSpace]", "-", - RowBox[{"0.004999588640440013`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.055989715330494454`", "\[VeryThinSpace]", "-", + RowBox[{"0.08886897373860805`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6427,8 +7585,8 @@ Cell[BoxData[ $CellContext`c[3]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.11702089598134102`", "\[VeryThinSpace]", "-", - RowBox[{"0.24889470697484575`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.0159420973600971`", "\[VeryThinSpace]", "+", + RowBox[{"0.0559517702136037`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6465,11 +7623,11 @@ Cell[BoxData[ $CellContext`c[3]], Q3`Dagger[ $CellContext`c[4]], - $CellContext`c[4]]]}], "+", + $CellContext`c[4]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.015346574282136729`", "\[VeryThinSpace]", "-", - RowBox[{"0.046478036035658865`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.15422284734529546`", "\[VeryThinSpace]", "+", + RowBox[{"0.07075117410663456`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6502,11 +7660,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[4]], $CellContext`c[2], - $CellContext`c[1]]]}], "-", + $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.09345606119665524`", "\[VeryThinSpace]", "-", - RowBox[{"0.01005390186762969`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.04123411019027247`", "\[VeryThinSpace]", "+", + RowBox[{"0.003986299740907147`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6539,11 +7697,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[4]], $CellContext`c[3], - $CellContext`c[1]]]}], "+", + $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.025133446049556588`", "\[VeryThinSpace]", "+", - RowBox[{"0.02069762483942861`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.052559094230861116`", "\[VeryThinSpace]", "-", + RowBox[{"0.10366645738290811`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6576,11 +7734,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[4]], $CellContext`c[3], - $CellContext`c[2]]]}], "-", + $CellContext`c[2]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.1844365142207739`", "\[VeryThinSpace]", "-", - RowBox[{"0.167237550185884`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.11933686428026469`", "\[VeryThinSpace]", "+", + RowBox[{"0.03655527302888932`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6616,8 +7774,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.05362077447895421`", "\[VeryThinSpace]", "+", - RowBox[{"0.06389316178723631`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.022451698571238635`", "\[VeryThinSpace]", "-", + RowBox[{"0.0013502430684809385`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6650,11 +7808,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[4]], $CellContext`c[4], - $CellContext`c[2]]]}], "+", + $CellContext`c[2]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.12852270321886883`", "\[VeryThinSpace]", "+", - RowBox[{"0.010133820640769772`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.033289319584911006`", "\[VeryThinSpace]", "-", + RowBox[{"0.08281103255654987`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6687,11 +7845,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[4]], $CellContext`c[4], - $CellContext`c[3]]]}], "-", + $CellContext`c[3]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.0073904611040086325`", "\[VeryThinSpace]", "+", - RowBox[{"0.01426656800389743`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.009650304819368578`", "\[VeryThinSpace]", "+", + RowBox[{"0.04092548203814837`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6723,8 +7881,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.06720188120105386`", "\[VeryThinSpace]", "-", - RowBox[{"0.09358902821529747`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.0017061095509202512`", "\[VeryThinSpace]", "+", + RowBox[{"0.09106484086442057`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6756,8 +7914,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.1324496701694546`", "\[VeryThinSpace]", "-", - RowBox[{"0.17186154280323107`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.028539475247507058`", "\[VeryThinSpace]", "+", + RowBox[{"0.059338086870217605`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6789,8 +7947,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.07334447243862642`", "\[VeryThinSpace]", "-", - RowBox[{"0.015244729427396707`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.05836807624483441`", "\[VeryThinSpace]", "-", + RowBox[{"0.010313153347727613`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6819,11 +7977,11 @@ Cell[BoxData[ $CellContext`c[3]], $CellContext`c[4], $CellContext`c[3], - $CellContext`c[2]]]}], "+", + $CellContext`c[2]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.08628095561219659`", "\[VeryThinSpace]", "-", - RowBox[{"0.016857428933747625`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.18783052637532788`", "\[VeryThinSpace]", "-", + RowBox[{"0.050377100747409`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6856,11 +8014,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[4]], $CellContext`c[2], - $CellContext`c[1]]]}], "-", + $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.17343323947702302`", "\[VeryThinSpace]", "-", - RowBox[{"0.00979687508200861`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.07544575394775431`", "\[VeryThinSpace]", "-", + RowBox[{"0.09710265740975682`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6896,8 +8054,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.06533036504476969`", "\[VeryThinSpace]", "+", - RowBox[{"0.019597555388709987`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.06972192335799499`", "\[VeryThinSpace]", "+", + RowBox[{"0.11559385302377215`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6930,11 +8088,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[4]], $CellContext`c[3], - $CellContext`c[2]]]}], "-", + $CellContext`c[2]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.07266220219893622`", "\[VeryThinSpace]", "-", - RowBox[{"0.11708579029985985`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.06611950383519799`", "\[VeryThinSpace]", "+", + RowBox[{"0.030777027687178413`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -6970,8 +8128,8 @@ Cell[BoxData[ $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.11970830046485419`", "\[VeryThinSpace]", "+", - RowBox[{"0.07275014600381256`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.038828294739620475`", "\[VeryThinSpace]", "+", + RowBox[{"0.008271808535887982`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -7007,8 +8165,8 @@ Cell[BoxData[ $CellContext`c[2]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.30696262638942795`", "\[VeryThinSpace]", "+", - RowBox[{"0.12046541353507004`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.014916980848278905`", "\[VeryThinSpace]", "+", + RowBox[{"0.03936558339715229`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -7041,11 +8199,11 @@ Cell[BoxData[ Q3`Dagger[ $CellContext`c[4]], $CellContext`c[4], - $CellContext`c[3]]]}], "-", + $CellContext`c[3]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.045474492425620044`", "\[VeryThinSpace]", "-", - RowBox[{"0.04491933070013626`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.04476822442750824`", "\[VeryThinSpace]", "-", + RowBox[{"0.06969089618137865`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -7074,11 +8232,11 @@ Cell[BoxData[ $CellContext`c[4]], $CellContext`c[3], $CellContext`c[2], - $CellContext`c[1]]]}], "+", + $CellContext`c[1]]]}], "-", RowBox[{ RowBox[{"(", - RowBox[{"0.0976838641116477`", "\[VeryThinSpace]", "+", - RowBox[{"0.13737242494447013`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.07731550914902581`", "\[VeryThinSpace]", "-", + RowBox[{"0.08265599044033872`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ TemplateBox[{ TemplateBox[{"\[VeryThinSpace]", "\"\[VeryThinSpace]\"", @@ -7107,11 +8265,11 @@ Cell[BoxData[ $CellContext`c[4]], $CellContext`c[4], $CellContext`c[2], - $CellContext`c[1]]]}], "-", + $CellContext`c[1]]]}], "+", RowBox[{ RowBox[{"(", - RowBox[{"0.1511856938730285`", 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0., {3, 7} -> + 0., {3, 8} -> 0., {4, 5} -> Complex[1.0000000000000002`, 0.], {4, 6} -> + 0., {4, 7} -> Complex[0.5970585731629219, -0.2878328789990047], {4, 8} -> + Complex[0.009986162210116247, -0.23371420625605585`], {5, 6} -> + 0., {5, 7} -> 0., {5, 8} -> 0., {6, 7} -> + Complex[0.3470646995742765, 0.49009335690727246`], {6, 8} -> + Complex[0.21892124648388725`, -0.019955878673239308`]}, Antisymmetric[{1, 2}]}]], { $CellContext`c[1], $CellContext`c[2], @@ -9173,8 +10305,8 @@ cKnHlQ5JtXegwgFXeYhLPa78RS334AoHACR45TI= Editable->False, SelectWithContents->True, Selectable->False]], "Output", - CellLabel->"Out[6]=", - CellID->1473765303,ExpressionUUID->"11b46c73-3d8c-40ad-aec4-f4966369a8e6"] + CellLabel->"Out[8]=", + CellID->334503293,ExpressionUUID->"e04bf97e-d0c1-4270-96a2-0c3a87d2b43d"] }, Open ]], Cell["We will take the expectation value of this operator.", "ExampleText", @@ -9189,7 +10321,7 @@ Cell[BoxData[ RowBox[{"c", "[", "3", "]"}], "]"}], "**", RowBox[{"c", "[", "1", 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CellID->1576300494,ExpressionUUID->"4613eb94-695a-40bd-aeef-da91ed3ec39a"] + RowBox[{"0.32299645825451445`", "\[VeryThinSpace]", "+", + RowBox[{"0.041700076635530575`", " ", "\[ImaginaryI]"}]}]], "Output", + CellLabel->"Out[10]=", + CellID->1776243266,ExpressionUUID->"91c91606-d299-4914-a827-3cf920367e28"] }, Open ]], Cell["\<\ @@ -9264,7 +10396,7 @@ Cell[BoxData[{ RowBox[{"cc", "->", RowBox[{"U2", ".", "U1", ".", "cc"}]}], "]"}]}]}], "Input", CellProlog:>Needs["Q3`"], - CellLabel->"In[9]:=", + CellLabel->"In[11]:=", CellID->811619388,ExpressionUUID->"0831df7c-f82e-40cc-acd0-cf27f2b58b9f"], Cell[BoxData[ @@ -9279,8 +10411,10 @@ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"(", - RowBox[{"0.2850552905056367`", "\[VeryThinSpace]", "+", - RowBox[{"0.330819776459948`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{ + RowBox[{"-", "0.36033783310782486`"}], "-", + RowBox[{"0.17567162691109262`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + InterpretationBox[ SubscriptBox["c", TemplateBox[{",", 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"\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"0.36786000710711714`", "\[VeryThinSpace]", "+", + RowBox[{"0.4304297410371871`", " ", "\[ImaginaryI]"}]}], ")"}], " ", InterpretationBox[ SubscriptBox["c", TemplateBox[{",", "\",\"", "4"}, "RowWithSeparators"]], $CellContext`c[4]]}]}]}]}], "}"}]], "Output", - CellLabel->"Out[9]=", - CellID->1569236731,ExpressionUUID->"23d07e98-251d-4f56-8ee7-ec283b23d072"], + CellLabel->"Out[11]=", + CellID->509356299,ExpressionUUID->"9fadaacb-e544-4bd1-bef0-e6525d5a9687"], Cell[BoxData[ RowBox[{"{", @@ -9467,18 +10599,18 @@ Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{ - RowBox[{"-", "0.451479886221137`"}], "-", - RowBox[{"0.3951660665389056`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + RowBox[{"-", "0.150958056811666`"}], "-", + RowBox[{"0.20829110878142082`", " ", "\[ImaginaryI]"}]}], ")"}], " ", + InterpretationBox[ SubscriptBox["c", TemplateBox[{",", "\",\"", "1"}, "RowWithSeparators"]], - $CellContext`c[1]]}], "-", + $CellContext`c[1]]}], "+", RowBox[{ 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a/Q3/Documentation/English/ReferencePages/Symbols/WickGreensFunction.nb b/Q3/Documentation/English/ReferencePages/Symbols/WickGreensFunction.nb index 59de780a1..ef47d9a8a 100644 --- a/Q3/Documentation/English/ReferencePages/Symbols/WickGreensFunction.nb +++ b/Q3/Documentation/English/ReferencePages/Symbols/WickGreensFunction.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 236123, 4969] -NotebookOptionsPosition[ 226089, 4770] -NotebookOutlinePosition[ 228793, 4834] -CellTagsIndexPosition[ 228669, 4828] +NotebookDataLength[ 269651, 5599] +NotebookOptionsPosition[ 259738, 5402] +NotebookOutlinePosition[ 262442, 5466] +CellTagsIndexPosition[ 262318, 5460] WindowTitle->WickGreensFunction WindowFrame->Normal*) @@ -27,11 +27,11 @@ Cell[BoxData[GridBox[{ TemplateBox[{12}, "Spacer1"], Cell["Q3 SYMBOL", "PacletNameCell", TextAlignment->Center,ExpressionUUID-> - "ac593fae-6c01-420e-92bd-8ad824aa6a2b"], + 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"aaf5bad7-46f2-4f27-96e5-ecf760943bed"], + "be56b13a-502f-4bb2-86b3-05e1e16883d6"], TemplateBox[{8}, "Spacer1"]}]], TextAlignment->Center,ExpressionUUID-> - "18c4e803-d58e-477d-885c-858710b9accf"], + "61b780b4-7b64-444e-a2ee-35c092762449"], Background->RGBColor[0.490196, 0.576471, 0.690196], ItemSize->Full], ""} }, @@ -54,8 +54,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "2138652f-db9e-4cf5-9890-affc281b1144"] - }],ExpressionUUID->"8f127dce-c567-471f-8a60-e641d7ba2c21"], + "a410a3ec-7b42-4114-b632-68b7522713b7"] + }],ExpressionUUID->"ed8790c2-aa20-47a7-a8b4-c832f27b8da7"], StripOnInput->False],{ "\"WickState\"" :> Documentation`HelpLookup["paclet:Q3/ref/WickState"], @@ -68,13 +68,17 @@ Cell[BoxData[GridBox[{ "\"WickGreensFunction\"" :> Documentation`HelpLookup["paclet:Q3/ref/WickGreensFunction"], "\"Measurement\"" :> - Documentation`HelpLookup["paclet:Q3/ref/Measurement"]}, + Documentation`HelpLookup["paclet:Q3/ref/Measurement"], + "\"NambuMatrix\"" :> + Documentation`HelpLookup["paclet:Q3/ref/NambuMatrix"], + "\"NambuRandomMatrix\"" :> + Documentation`HelpLookup["paclet:Q3/ref/NambuRandomMatrix"]}, Appearance->None, MenuAppearance->Automatic, MenuStyle->"SeeAlso"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "aacbaf54-7c51-4fb8-a487-dfb31ce90fd2"], + "e9a1f764-61ac-421d-847c-0e5d1903f54a"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -93,8 +97,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "c1da191d-fc8a-4470-897d-7dff20c2e17e"] - }],ExpressionUUID->"39e17773-1a09-4626-aa2e-78cfb49bd71d"], + "c8f9ef00-8c40-47e4-b734-f2e4c114a47c"] + }],ExpressionUUID->"9d3912f3-ecf2-4c17-899a-9f13b4b60db1"], StripOnInput->False],{ "\"Quantum Many-Body Systems\"" :> Documentation`HelpLookup["paclet:Q3/guide/QuantumManyBodySystems"], @@ -104,7 +108,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"MoreAbout"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "cdd9bd22-ac00-49d2-a313-400b8d82bc54"], + "3f968448-5efd-47e4-87de-232dd87a5621"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -123,8 +127,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "9a4347e2-4be7-4df1-b966-f15b3bc7f187"] - }],ExpressionUUID->"b9a85834-2be3-4119-b7b7-a62534242e19"], + "46294182-f587-411a-a1a4-1cab1a251a2b"] + }],ExpressionUUID->"b2e3882c-a67c-4013-8f37-eb460b15cf93"], StripOnInput->False],{ "\"Quantum Many-Body Systems with Q3\"" :> Documentation`HelpLookup[ @@ -135,7 +139,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"Tutorials"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "ea985797-09df-44da-bf61-361acb05cb02"], + "e2c85017-ce29-4e1d-baa4-c9fe2e3bf6aa"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -154,8 +158,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "de9b050c-4796-4292-b63f-cddcdf66d160"] - }],ExpressionUUID->"6cef7a30-bd34-4ebc-9d81-4b0e605c9f3c"], + "e00b6374-e5f7-4609-aa5a-9805e2f3c0b8"] + }],ExpressionUUID->"52da6bcd-2121-454a-8e20-0ef8cda78b83"], StripOnInput->False],{ "\"Q3/ref/WickRandomCircuit\"" :> None, "\"Copy Wolfram Documentation Center URL\"" :> @@ -192,14 +196,14 @@ WickRandomCircuit.html"], StandardForm]], "Input", TextClipboardType -> MenuStyle->"URLMenu"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - 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"41e0c0eb-f6ff-4569-a3e9-060a2f58b73a"], "\"\\"", + "ad30c944-48a0-4fc1-8b3f-38876ae0c88d"], "\"\\"", "Automatic", Cell["\<\ A list of 0 or 1 specifying the initial occupation of the fermion modes. \ Defaulti is the half-filling.\ \>", "TableText",ExpressionUUID->"94ea78a9-146a-481f-a33a-a03d3895c4b2"]}, {Cell[" ", "TableRowIcon",ExpressionUUID-> - "6b34b355-72ed-482b-b0ce-709b46f86b90"], "\"\\"", + "17a1a2df-c0f5-4221-af88-eacd4b7d3801"], "\"\\"", RowBox[{"{", RowBox[{"10", ",", " ", "5"}], "}"}], Cell[TextData[{ Cell[BoxData[ @@ -387,7 +391,7 @@ Defaulti is the half-filling.\ " times." }], "TableText",ExpressionUUID->"ef55ff0a-b2d0-4963-a42e-a2f2e191d946"]}, {Cell[" ", "TableRowIcon",ExpressionUUID-> - "2805b672-1355-4d9d-8972-172b1ae63c72"], "\"\\"", + "a6c12ac1-88b0-4202-b47f-2747aa38d1b6"], "\"\\"", TemplateBox[{ Cell[ TextData["False"]], "paclet:ref/False"}, @@ -396,7 +400,7 @@ Defaulti is the half-filling.\ "If True, the trajectories are saved to a file.", "TableText", ExpressionUUID->"7435d185-cd25-4673-b761-df92726f404c"]}, {Cell[" ", "TableRowIcon",ExpressionUUID-> - 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Complex[-0.3239368353423375, -0.27853835234466173`], {6, + 12} -> Complex[-0.19299966150123138`, + 0.3282026155242766], {6, 13} -> + Complex[-0.3239368353423375, -0.27853835234466173`], {6, + 14} -> Complex[-0.19299966150123138`, + 0.3282026155242766], {6, 15} -> + Complex[0.13476314496413244`, 0.0767423516894553], {6, + 16} -> Complex[ + 0.12289396168675226`, -0.38209796458606343`], {6, 17} -> + Complex[0.2578656808252187, -0.10876030801336257`], {6, + 18} -> Complex[-0.17834040611966948`, \ +-0.12609948481270494`], {6, 19} -> Complex[0.7835951260215941, 0.], {6, 20} -> + Complex[ + 8.326672684688674*^-17, -1.3877787807814457`*^-17], {6, + 21} -> Complex[-0.5748910514001031, \ +-0.14633045383134305`], {6, 22} -> + Complex[-0.15211930857597808`, -0.0462428201376506], {6, + 23} -> Complex[ + 0.029019428968993538`, 0.21528502386671927`], {6, 24} -> + Complex[0.46472596940822686`, 0.06199143840307167], {7, + 8} -> Complex[0.47738623440613004`, 0.], {7, 9} -> + Complex[0.08485061223426596, 0.11980477813807357`], {7, + 10} -> Complex[-0.17479296155780177`, + 0.14732776197358502`], {7, 11} -> + Complex[0.09710709374348331, 0.20541302638200343`], {7, + 12} -> Complex[-0.04249050665712762, -0.090268967382963], \ +{7, 13} -> Complex[0.09710709374348331, 0.20541302638200343`], {7, 14} -> + Complex[-0.04249050665712762, -0.090268967382963], {7, + 15} -> Complex[ + 0.08485061223426596, 0.11980477813807357`], {7, 16} -> + Complex[-0.17479296155780177`, 0.14732776197358502`], {7, + 17} -> Complex[-1.7867651802561116`*^-16, \ +-1.1449174941446927`*^-16], {7, 18} -> + Complex[0.47738623440613004`, 0.], {7, 19} -> + Complex[-0.17834040611966948`, 0.12609948481270494`], {7, + 20} -> Complex[ + 0.22516518597054483`, 0.1013577129442031], {7, 21} -> + Complex[0.2616855917631509, -0.009667937264448562], {7, + 22} -> Complex[ + 0.053161902759652525`, -0.14451247653181823`], {7, 23} -> + Complex[-0.16418162796222252`, 0.13133580222113328`], {7, + 24} -> Complex[-0.42262163352679005`, + 0.31456841745062747`], {8, 9} -> + Complex[0.17479296155780188`, 0.14732776197358494`], {8, + 10} -> Complex[-0.08485061223426603, + 0.11980477813807351`], {8, 11} -> + Complex[-0.13183382282185513`, -0.15967490110950291`], {8, + 12} -> Complex[ + 0.11065480592987563`, -0.022498803837048064`], {8, 13} -> + Complex[-0.13183382282185513`, -0.15967490110950291`], {8, + 14} -> Complex[ + 0.11065480592987563`, -0.022498803837048064`], {8, 15} -> + Complex[0.17479296155780188`, 0.14732776197358494`], {8, + 16} -> Complex[-0.08485061223426603, + 0.11980477813807351`], {8, 17} -> + Complex[0.5226137655938699, 0.], {8, 18} -> + Complex[-1.7867651802561116`*^-16, + 1.1449174941446927`*^-16], {8, 19} -> + Complex[0.2578656808252187, 0.10876030801336257`], {8, + 20} -> Complex[-0.007671097858546656, + 0.0057340892009139326`], {8, 21} -> + Complex[0.037739555243020126`, -0.012308050579545225`], { + 8, 22} -> + Complex[-0.13167457410588743`, -0.003054508743256395], {8, + 23} -> + Complex[-0.23097227051538433`, -0.42320695740719316`], {8, + 24} -> Complex[ + 0.29687889692943875`, 0.2375326567068352], {9, 10} -> + Complex[0.619974902703056, 0.], {9, 11} -> + Complex[0.13800495411149877`, -0.3680867713233184], {9, + 12} -> Complex[-0.1905043599805688, \ +-0.13762822610920328`], {9, 13} -> + Complex[0.13800495411149877`, -0.3680867713233184], {9, + 14} -> Complex[-0.1905043599805688, \ +-0.13762822610920328`], {9, 15} -> + Complex[-8.673617379884035*^-17, + 8.326672684688674*^-17], {9, 16} -> + Complex[0.619974902703056, 0.], {9, 17} -> + Complex[-0.08485061223426603, -0.11980477813807351`], {9, + 18} -> Complex[-0.17479296155780177`, \ +-0.14732776197358502`], {9, 19} -> + Complex[0.12289396168675226`, 0.38209796458606343`], {9, + 20} -> Complex[-0.09460977915945387, \ +-0.11933166053315665`], {9, 21} -> + Complex[-0.2346066684020043, -0.5335785389713812], {9, + 22} -> Complex[ + 0.08210404404737587, 0.05307307167276523], {9, 23} -> + Complex[-0.2917595352077467, 0.3784536838475334], {9, 24} -> + Complex[0.33333532926317844`, 0.0052698415932301845`], { + 10, 11} -> + Complex[-0.15558917647409173`, -0.09979098803090053], {10, + 12} -> Complex[ + 0.1899783589314758, -0.1365082746823964], {10, 13} -> + Complex[-0.15558917647409173`, -0.09979098803090053], {10, + 14} -> Complex[ + 0.1899783589314758, -0.1365082746823964], {10, 15} -> + Complex[0.3800250972969439, 0.], {10, 16} -> + Complex[-8.673617379884035*^-17, \ +-8.326672684688674*^-17], {10, 17} -> + Complex[0.17479296155780188`, -0.14732776197358494`], {10, + 18} -> Complex[ + 0.08485061223426596, -0.11980477813807357`], {10, 19} -> + Complex[0.13476314496413244`, -0.0767423516894553], {10, + 20} -> Complex[ + 0.02081012384838236, -0.14234469229568147`], {10, 21} -> + Complex[-0.15157725194981503`, 0.17844414940708186`], {10, + 22} -> + Complex[-0.04368290380765367, -0.027436459786564067`], { + 10, 23} -> + Complex[0.01747597846754094, 0.10203270358947614`], {10, + 24} -> Complex[ + 0.34087862056969404`, 0.1100388283919505], {11, 12} -> + Complex[0.5275300911030236, 0.], {11, 13} -> + Complex[1.8041124150158796`*^-16, + 1.457167719820518*^-16], {11, 14} -> + Complex[0.5275300911030236, 0.], {11, 15} -> + Complex[0.1899783589314758, 0.1365082746823964], {11, 16} -> + Complex[-0.1905043599805688, 0.13762822610920328`], {11, + 17} -> Complex[ + 0.11065480592987563`, 0.022498803837048064`], {11, 18} -> + Complex[-0.04249050665712762, 0.090268967382963], {11, + 19} -> Complex[-0.19299966150123138`, \ +-0.3282026155242766], {11, 20} -> + Complex[0.017368709836144688`, -0.1497980533645717], {11, + 21} -> Complex[-0.06250476454855013, + 0.34969666557353624`], {11, 22} -> + Complex[-0.008146862343863803, 0.10273781027166022`], {11, + 23} -> Complex[ + 0.11666092373220538`, -0.218936695047531], {11, 24} -> + Complex[0.161016937967343, 0.024863370726720582`], {12, + 13} -> Complex[0.4724699088969763, 0.], {12, 14} -> + Complex[ + 1.8041124150158796`*^-16, -1.457167719820518*^-16], {12, + 15} -> Complex[-0.15558917647409173`, + 0.09979098803090053], {12, 16} -> + Complex[0.13800495411149877`, 0.3680867713233184], {12, + 17} -> Complex[-0.13183382282185513`, + 0.15967490110950291`], {12, 18} -> + Complex[0.09710709374348331, -0.20541302638200343`], {12, + 19} -> Complex[-0.3239368353423375, + 0.27853835234466173`], {12, 20} -> + Complex[0.13688251203715338`, -0.1036078618111689], {12, 21} -> Complex[ - 0.07416506719537758, 0.08051189278847616], {8, 22} -> - Complex[-0.07115780937991523, -0.011205895451017422`], {8, + 0.25833102475550235`, -0.37839808713437206`], {12, 22} -> + Complex[0.01476498538421066, -0.0004105289599095289], {12, + 23} -> Complex[-0.0868392972934217, + 0.014513008624828502`], {12, 24} -> + Complex[-0.16585075463443222`, 0.4085141858532183], {13, + 14} -> Complex[0.5275300911030236, 0.], {13, 15} -> + Complex[0.1899783589314758, 0.1365082746823964], {13, 16} -> + Complex[-0.1905043599805688, 0.13762822610920328`], {13, + 17} -> Complex[ + 0.11065480592987563`, 0.022498803837048064`], {13, 18} -> + Complex[-0.04249050665712762, 0.090268967382963], {13, + 19} -> Complex[-0.19299966150123138`, \ +-0.3282026155242766], {13, 20} -> + Complex[0.017368709836144688`, -0.1497980533645717], {13, + 21} -> Complex[-0.06250476454855013, + 0.34969666557353624`], {13, 22} -> + Complex[-0.008146862343863803, 0.10273781027166022`], {13, 23} -> Complex[ - 0.16774613988004144`, 0.225655277072902], {8, 24} -> - Complex[-0.09714752480970831, -0.05391368675976281], {9, - 10} -> Complex[0.4159087185308232, 0.], {9, 11} -> - Complex[0.3689066714841379, -0.01963818646965646], {9, - 12} -> Complex[-0.364332505630415, \ --0.055629112414147225`], {9, 13} -> - Complex[0.3689066714841379, -0.01963818646965646], {9, - 14} -> Complex[-0.364332505630415, \ --0.055629112414147225`], {9, 15} -> - Complex[-0.29089675324323006`, 0.1558046632134409], {9, - 16} -> Complex[0.4159087185308232, 0.], {9, 17} -> - Complex[0.06911881436797135, -0.04269376157510484], {9, - 18} -> Complex[ - 0.14880448551485936`, 0.18343023977218464`], {9, 19} -> - Complex[0.0806509155236052, -0.06238441832735686], {9, - 20} -> Complex[-0.08181885113853848, - 0.0223150746441549], {9, 21} -> - Complex[0.011359633728347016`, -0.08520800008182422], {9, + 0.11666092373220538`, -0.218936695047531], {13, 24} -> + Complex[0.161016937967343, 0.024863370726720582`], {14, + 15} -> Complex[-0.15558917647409173`, + 0.09979098803090053], {14, 16} -> + Complex[0.13800495411149877`, 0.3680867713233184], {14, + 17} -> Complex[-0.13183382282185513`, + 0.15967490110950291`], {14, 18} -> + Complex[0.09710709374348331, -0.20541302638200343`], {14, + 19} -> Complex[-0.3239368353423375, + 0.27853835234466173`], {14, 20} -> + Complex[0.13688251203715338`, -0.1036078618111689], {14, + 21} -> Complex[ + 0.25833102475550235`, -0.37839808713437206`], {14, 22} -> + Complex[0.01476498538421066, -0.0004105289599095289], {14, + 23} -> Complex[-0.0868392972934217, + 0.014513008624828502`], {14, 24} -> + Complex[-0.16585075463443222`, 0.4085141858532183], {15, + 16} -> Complex[0.619974902703056, 0.], {15, 17} -> + Complex[-0.08485061223426603, -0.11980477813807351`], {15, + 18} -> + Complex[-0.17479296155780177`, -0.14732776197358502`], { + 15, 19} -> + Complex[0.12289396168675226`, 0.38209796458606343`], {15, + 20} -> Complex[-0.09460977915945387, \ +-0.11933166053315665`], {15, 21} -> + Complex[-0.2346066684020043, -0.5335785389713812], {15, 22} -> Complex[ - 0.3146513247958228, -0.026797378998685423`], {9, 23} -> - Complex[0.17837001887568327`, 0.09187489939370964], {9, - 24} -> Complex[-0.07164664525064873, - 0.2907134789193454], {10, 11} -> - Complex[-0.1634450723422762, -0.0011076658174966336`], { - 10, 12} -> - Complex[0.24390216997712094`, 0.12967162455970677`], {10, - 13} -> Complex[-0.1634450723422762, \ --0.0011076658174966336`], {10, 14} -> - Complex[0.24390216997712094`, 0.12967162455970677`], {10, - 15} -> Complex[0.41416215192864014`, 0.], {10, 16} -> - Complex[-0.29089675324323006`, -0.1558046632134409], {10, - 17} -> Complex[-0.11463533571593523`, \ --0.055210078698593804`], {10, 18} -> - Complex[0.1868664022204954, -0.06385028540854319], {10, - 19} -> Complex[-0.2934854494126851, \ --0.04085502210736608], {10, 20} -> - Complex[0.22168267144312054`, -0.03299920735716682], {10, - 21} -> Complex[-0.030346695869449408`, \ --0.030149558835644692`], {10, 22} -> - Complex[-0.06237372995183321, 0.06727056435677253], {10, - 23} -> Complex[-0.09727281618814543, \ --0.4251726947214757], {10, 24} -> - Complex[0.02023138254715448, -0.11137525641609], {11, 12} -> - Complex[0.3744935743083003, 0.], {11, 13} -> - Complex[-0.4009452410407828, 0.11284019268466222`], {11, - 14} -> Complex[0.3744935743083003, 0.], {11, 15} -> - Complex[0.24390216997712094`, -0.12967162455970677`], {11, - 16} -> Complex[-0.364332505630415, - 0.055629112414147225`], {11, 17} -> - Complex[-0.010533198723196371`, 0.006296053518233197], { - 11, 18} -> - Complex[-0.2300764357164763, -0.11137567802408056`], {11, - 19} -> Complex[-0.029186234486845222`, - 0.022372160811117286`], {11, 20} -> - Complex[0.14142597329959572`, 0.061350391806736695`], {11, - 21} -> Complex[ - 0.09059534447844704, 0.10345647613127328`], {11, 22} -> - Complex[-0.33684082460892883`, 0.09005315780013691], {11, - 23} -> Complex[-0.008851768247553363, \ --0.029068202106085572`], {11, 24} -> - Complex[-0.07795077274116374, -0.2904671042221048], {12, - 13} -> Complex[0.5794387931418044, 0.], {12, 14} -> - Complex[-0.4009452410407828, -0.11284019268466222`], {12, - 15} -> Complex[-0.1634450723422762, - 0.0011076658174966336`], {12, 16} -> - Complex[0.3689066714841379, 0.01963818646965646], {12, - 17} -> Complex[-0.06875971685231415, \ --0.006550328119388341], {12, 18} -> - Complex[0.4351121808870284, 0.1282880779202612], {12, 19} -> - Complex[-0.18425433538775948`, 0.03665899040288985], {12, - 20} -> Complex[-0.04214475287108173, \ --0.2125642839376449], {12, 21} -> - Complex[-0.11604982216622163`, -0.19926182212673743`], { - 12, 22} -> - Complex[0.5691291778553188, 0.009546455336403484], {12, - 23} -> Complex[-0.20476687273928368`, \ --0.20940927658482447`], {12, 24} -> - Complex[-0.04633762187626639, 0.37156097014999256`], {13, - 14} -> Complex[0.3744935743083003, 0.], {13, 15} -> - Complex[0.24390216997712094`, -0.12967162455970677`], {13, - 16} -> Complex[-0.364332505630415, - 0.055629112414147225`], {13, 17} -> - Complex[-0.010533198723196371`, 0.006296053518233197], { - 13, 18} -> - Complex[-0.2300764357164763, -0.11137567802408056`], {13, - 19} -> Complex[-0.029186234486845222`, - 0.022372160811117286`], {13, 20} -> - Complex[0.14142597329959572`, 0.061350391806736695`], {13, - 21} -> Complex[ - 0.09059534447844704, 0.10345647613127328`], {13, 22} -> - Complex[-0.33684082460892883`, 0.09005315780013691], {13, - 23} -> Complex[-0.008851768247553363, \ --0.029068202106085572`], {13, 24} -> - Complex[-0.07795077274116374, -0.2904671042221048], {14, - 15} -> Complex[-0.1634450723422762, - 0.0011076658174966336`], {14, 16} -> - Complex[0.3689066714841379, 0.01963818646965646], {14, - 17} -> Complex[-0.06875971685231415, \ --0.006550328119388341], {14, 18} -> - Complex[0.4351121808870284, 0.1282880779202612], {14, 19} -> - Complex[-0.18425433538775948`, 0.03665899040288985], {14, - 20} -> Complex[-0.04214475287108173, \ --0.2125642839376449], {14, 21} -> - Complex[-0.11604982216622163`, -0.19926182212673743`], { - 14, 22} -> - Complex[0.5691291778553188, 0.009546455336403484], {14, - 23} -> Complex[-0.20476687273928368`, \ --0.20940927658482447`], {14, 24} -> - Complex[-0.04633762187626639, 0.37156097014999256`], {15, - 16} -> Complex[0.4159087185308232, 0.], {15, 17} -> - Complex[0.06911881436797135, -0.04269376157510484], {15, - 18} -> Complex[ - 0.14880448551485936`, 0.18343023977218464`], {15, 19} -> - Complex[0.0806509155236052, -0.06238441832735686], {15, - 20} -> 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-0.4251726947214757], {10, 24} -> - Complex[0.02023138254715448, -0.11137525641609], {11, 12} -> - Complex[0.3744935743083003, 0.], {11, 13} -> - Complex[-0.4009452410407828, 0.11284019268466222`], {11, 14} -> - Complex[0.3744935743083003, 0.], {11, 15} -> - Complex[0.24390216997712094`, -0.12967162455970677`], {11, 16} -> - Complex[-0.364332505630415, 0.055629112414147225`], {11, 17} -> - Complex[-0.010533198723196371`, 0.006296053518233197], {11, 18} -> - Complex[-0.2300764357164763, -0.11137567802408056`], {11, 19} -> - Complex[-0.029186234486845222`, 0.022372160811117286`], {11, 20} -> - Complex[0.14142597329959572`, 0.061350391806736695`], {11, 21} -> - Complex[0.09059534447844704, 0.10345647613127328`], {11, 22} -> - Complex[-0.33684082460892883`, 0.09005315780013691], {11, 23} -> - Complex[-0.008851768247553363, -0.029068202106085572`], {11, 24} -> - Complex[-0.07795077274116374, -0.2904671042221048], {12, 13} -> - Complex[0.5794387931418044, 0.], {12, 14} -> - Complex[-0.4009452410407828, -0.11284019268466222`], {12, 15} -> - Complex[-0.1634450723422762, 0.0011076658174966336`], {12, 16} -> - Complex[0.3689066714841379, 0.01963818646965646], {12, 17} -> - Complex[-0.06875971685231415, -0.006550328119388341], {12, 18} -> - Complex[0.4351121808870284, 0.1282880779202612], {12, 19} -> - Complex[-0.18425433538775948`, 0.03665899040288985], {12, 20} -> - Complex[-0.04214475287108173, -0.2125642839376449], {12, 21} -> - Complex[-0.11604982216622163`, -0.19926182212673743`], {12, 22} -> - Complex[0.5691291778553188, 0.009546455336403484], {12, 23} -> - Complex[-0.20476687273928368`, -0.20940927658482447`], {12, 24} -> - Complex[-0.04633762187626639, 0.37156097014999256`], {13, 14} -> - Complex[0.3744935743083003, 0.], {13, 15} -> - Complex[0.24390216997712094`, -0.12967162455970677`], {13, 16} -> - Complex[-0.364332505630415, 0.055629112414147225`], {13, 17} -> - Complex[-0.010533198723196371`, 0.006296053518233197], {13, 18} -> - Complex[-0.2300764357164763, -0.11137567802408056`], {13, 19} -> - Complex[-0.029186234486845222`, 0.022372160811117286`], {13, 20} -> - Complex[0.14142597329959572`, 0.061350391806736695`], {13, 21} -> - Complex[0.09059534447844704, 0.10345647613127328`], {13, 22} -> - Complex[-0.33684082460892883`, 0.09005315780013691], {13, 23} -> - Complex[-0.008851768247553363, -0.029068202106085572`], {13, 24} -> - Complex[-0.07795077274116374, -0.2904671042221048], {14, 15} -> - Complex[-0.1634450723422762, 0.0011076658174966336`], {14, 16} -> - Complex[0.3689066714841379, 0.01963818646965646], {14, 17} -> - Complex[-0.06875971685231415, -0.006550328119388341], {14, 18} -> - Complex[0.4351121808870284, 0.1282880779202612], {14, 19} -> - Complex[-0.18425433538775948`, 0.03665899040288985], {14, 20} -> - Complex[-0.04214475287108173, -0.2125642839376449], {14, 21} -> - Complex[-0.11604982216622163`, -0.19926182212673743`], {14, 22} -> - Complex[0.5691291778553188, 0.009546455336403484], {14, 23} -> - Complex[-0.20476687273928368`, -0.20940927658482447`], {14, 24} -> - Complex[-0.04633762187626639, 0.37156097014999256`], {15, 16} -> - Complex[0.4159087185308232, 0.], {15, 17} -> - Complex[0.06911881436797135, -0.04269376157510484], {15, 18} -> - Complex[0.14880448551485936`, 0.18343023977218464`], {15, 19} -> - Complex[0.0806509155236052, -0.06238441832735686], {15, 20} -> - Complex[-0.08181885113853848, 0.0223150746441549], {15, 21} -> - Complex[0.011359633728347016`, -0.08520800008182422], {15, 22} -> - Complex[0.3146513247958228, -0.026797378998685423`], {15, 23} -> - Complex[0.17837001887568327`, 0.09187489939370964], {15, 24} -> - Complex[-0.07164664525064873, 0.2907134789193454], {16, 17} -> - Complex[-0.11463533571593523`, -0.055210078698593804`], {16, 18} -> - Complex[0.1868664022204954, -0.06385028540854319], {16, 19} -> - Complex[-0.2934854494126851, -0.04085502210736608], {16, 20} -> - Complex[0.22168267144312054`, -0.03299920735716682], {16, 21} -> - Complex[-0.030346695869449408`, -0.030149558835644692`], {16, 22} -> - Complex[-0.06237372995183321, 0.06727056435677253], {16, 23} -> - Complex[-0.09727281618814543, -0.4251726947214757], {16, 24} -> - Complex[0.02023138254715448, -0.11137525641609], {17, 18} -> - Complex[0.7694387170582185, 0.], {17, 19} -> - Complex[-0.4024303768625533, -0.06319936655428628], {17, 20} -> - Complex[-0.02707098756043888, -0.08415309369657159], {17, 21} -> - Complex[-0.2385656438458264, -0.06903700176549488], {17, 22} -> - Complex[0.6164020052722983, 0.03301684548082712], {17, 23} -> - Complex[-0.24737746704016716`, -0.5022802985841684], {17, 24} -> - Complex[0.199408104684111, 0.11720650121625686`], {18, 19} -> - Complex[0.12702736104247997`, -0.01646149694625006], {18, 20} -> - Complex[-0.03764992141799037, 0.13605099778794436`], {18, 21} -> - Complex[0.07416506719537758, 0.08051189278847616], {18, 22} -> - Complex[-0.07115780937991523, -0.011205895451017422`], {18, 23} -> - Complex[0.16774613988004144`, 0.225655277072902], {18, 24} -> - Complex[-0.09714752480970831, -0.05391368675976281], {19, 20} -> - Complex[0.460087813322552, 0.], {19, 21} -> - Complex[0.24014852879671952`, -0.15329539227786965`], {19, 22} -> - Complex[-0.020878723063805726`, 0.1044611112500656], {19, 23} -> - Complex[0.11519091687891489`, -0.4012928958026726], {19, 24} -> - Complex[-0.45588071484204273`, 0.014752170920114424`], {20, 21} -> - Complex[0.02087872306380592, 0.10446111125006569`], {20, 22} -> - Complex[-0.32198385415100245`, -0.056214448752933965`], {20, 23} -> - Complex[0.34202714802636813`, 0.5203828231210746], {20, 24} -> - Complex[0.27130379014783124`, 0.12044715596755934`], {21, 22} -> - Complex[0.7322837744378387, 0.], {21, 23} -> - Complex[-0.29340373279675197`, -0.27809938962030384`], {21, 24} -> - Complex[-0.07968743240634073, 0.17702649633258472`], {22, 23} -> - Complex[0.27244326195988916`, -0.011111878577649018`], {22, 24} -> - Complex[-0.3506829757791454, -0.10726001244941838`]}, + Complex[-0.2782895722096578, 0.5061420791848172], {2, 4} -> + Complex[-0.06947320790243908, -0.12537107773431694`], {2, 5} -> + Complex[0.029019428968993538`, -0.21528502386671927`], {2, 6} -> + Complex[-0.06797212261961828, -0.14596193524618625`], {2, 7} -> + Complex[-0.23097227051538433`, 0.42320695740719316`], {2, 8} -> + Complex[-0.16418162796222252`, -0.13133580222113328`], {2, 9} -> + Complex[0.01747597846754094, -0.10203270358947614`], {2, 10} -> + Complex[-0.2917595352077467, -0.3784536838475334], {2, 11} -> + Complex[-0.0868392972934217, -0.014513008624828502`], {2, 12} -> + Complex[0.11666092373220538`, 0.218936695047531], {2, 13} -> + Complex[-0.0868392972934217, -0.014513008624828502`], {2, 14} -> + Complex[0.11666092373220538`, 0.218936695047531], {2, 15} -> + Complex[0.01747597846754094, -0.10203270358947614`], {2, 16} -> + Complex[-0.2917595352077467, -0.3784536838475334], {2, 17} -> + Complex[-0.23097227051538433`, 0.42320695740719316`], {2, 18} -> + Complex[-0.16418162796222252`, -0.13133580222113328`], {2, 19} -> + Complex[0.029019428968993538`, -0.21528502386671927`], {2, 20} -> + Complex[-0.06797212261961828, -0.14596193524618625`], {2, 21} -> + Complex[-0.2782895722096578, 0.5061420791848172], {2, 22} -> + Complex[-0.06947320790243908, -0.12537107773431694`], {2, 23} -> + 1, {3, 4} -> Complex[0.12373001666450631`, 0.], {3, 5} -> + Complex[-0.15211930857597808`, 0.0462428201376506], {3, 6} -> + Complex[-0.029720518023860276`, 0.053544088868121745`], {3, 7} -> + Complex[-0.13167457410588743`, 0.003054508743256395], {3, 8} -> + Complex[0.053161902759652525`, 0.14451247653181823`], {3, 9} -> + Complex[-0.04368290380765367, 0.027436459786564067`], {3, 10} -> + Complex[0.08210404404737587, -0.05307307167276523], {3, 11} -> + Complex[0.01476498538421066, 0.0004105289599095289], {3, 12} -> + Complex[-0.008146862343863803, -0.10273781027166022`], {3, 13} -> + Complex[0.01476498538421066, 0.0004105289599095289], {3, 14} -> + Complex[-0.008146862343863803, -0.10273781027166022`], {3, 15} -> + Complex[-0.04368290380765367, 0.027436459786564067`], {3, 16} -> + Complex[0.08210404404737587, -0.05307307167276523], {3, 17} -> + Complex[-0.13167457410588743`, 0.003054508743256395], {3, 18} -> + Complex[0.053161902759652525`, 0.14451247653181823`], {3, 19} -> + Complex[-0.15211930857597808`, 0.0462428201376506], {3, 20} -> + Complex[-0.029720518023860276`, 0.053544088868121745`], {3, 21} -> + Complex[-1.1102230246251565`*^-16, 0.], {3, 22} -> + Complex[0.12373001666450631`, 0.], {3, 23} -> + Complex[-0.06947320790243908, 0.12537107773431694`], {3, 24} -> + Complex[-0.1766768302665536, -0.1352957631030186], {4, 5} -> + Complex[-0.5748910514001031, 0.14633045383134305`], {4, 6} -> + Complex[0.030636763457055183`, 0.11355874224143539`], {4, 7} -> + Complex[0.037739555243020126`, 0.012308050579545225`], {4, 8} -> + Complex[0.2616855917631509, 0.009667937264448562], {4, 9} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {4, 10} -> + Complex[-0.2346066684020043, 0.5335785389713812], {4, 11} -> + Complex[0.25833102475550235`, 0.37839808713437206`], {4, 12} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {4, 13} -> + Complex[0.25833102475550235`, 0.37839808713437206`], {4, 14} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {4, 15} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {4, 16} -> + Complex[-0.2346066684020043, 0.5335785389713812], {4, 17} -> + Complex[0.037739555243020126`, 0.012308050579545225`], {4, 18} -> + Complex[0.2616855917631509, 0.009667937264448562], {4, 19} -> + Complex[-0.5748910514001031, 0.14633045383134305`], {4, 20} -> + Complex[0.030636763457055183`, 0.11355874224143539`], {4, 21} -> + Complex[0.8762699833354937, 0.], {4, 22} -> + Complex[-1.1102230246251565`*^-16, 0.], {4, 23} -> + Complex[-0.2782895722096578, -0.5061420791848172], {4, 24} -> + Complex[-0.41976082253431385`, 0.08596477743313696], {5, 6} -> + Complex[0.21640487397840627`, 0.], {5, 7} -> + Complex[-0.007671097858546656, -0.0057340892009139326`], {5, 8} -> + Complex[0.22516518597054483`, -0.1013577129442031], {5, 9} -> + Complex[0.02081012384838236, 0.14234469229568147`], {5, 10} -> + Complex[-0.09460977915945387, 0.11933166053315665`], {5, 11} -> + Complex[0.13688251203715338`, 0.1036078618111689], {5, 12} -> + Complex[0.017368709836144688`, 0.1497980533645717], {5, 13} -> + Complex[0.13688251203715338`, 0.1036078618111689], {5, 14} -> + Complex[0.017368709836144688`, 0.1497980533645717], {5, 15} -> + Complex[0.02081012384838236, 0.14234469229568147`], {5, 16} -> + Complex[-0.09460977915945387, 0.11933166053315665`], {5, 17} -> + Complex[-0.007671097858546656, -0.0057340892009139326`], {5, 18} -> + Complex[0.22516518597054483`, -0.1013577129442031], {5, 19} -> + Complex[8.326672684688674*^-17, 1.3877787807814457`*^-17], {5, 20} -> + Complex[0.21640487397840627`, 0.], {5, 21} -> + Complex[0.030636763457055183`, -0.11355874224143539`], {5, 22} -> + Complex[-0.029720518023860276`, -0.053544088868121745`], {5, 23} -> + Complex[-0.06797212261961828, 0.14596193524618625`], {5, 24} -> + Complex[-0.1599463611986579, 0.3186964622408383], {6, 7} -> + Complex[0.2578656808252187, -0.10876030801336257`], {6, 8} -> + Complex[-0.17834040611966948`, -0.12609948481270494`], {6, 9} -> + Complex[0.13476314496413244`, 0.0767423516894553], {6, 10} -> + Complex[0.12289396168675226`, -0.38209796458606343`], {6, 11} -> + Complex[-0.3239368353423375, -0.27853835234466173`], {6, 12} -> + Complex[-0.19299966150123138`, 0.3282026155242766], {6, 13} -> + Complex[-0.3239368353423375, -0.27853835234466173`], {6, 14} -> + Complex[-0.19299966150123138`, 0.3282026155242766], {6, 15} -> + Complex[0.13476314496413244`, 0.0767423516894553], {6, 16} -> + Complex[0.12289396168675226`, -0.38209796458606343`], {6, 17} -> + Complex[0.2578656808252187, -0.10876030801336257`], {6, 18} -> + Complex[-0.17834040611966948`, -0.12609948481270494`], {6, 19} -> + Complex[0.7835951260215941, 0.], {6, 20} -> + Complex[8.326672684688674*^-17, -1.3877787807814457`*^-17], {6, + 21} -> Complex[-0.5748910514001031, -0.14633045383134305`], {6, + 22} -> Complex[-0.15211930857597808`, -0.0462428201376506], {6, + 23} -> Complex[0.029019428968993538`, 0.21528502386671927`], {6, + 24} -> Complex[0.46472596940822686`, 0.06199143840307167], {7, 8} -> + Complex[0.47738623440613004`, 0.], {7, 9} -> + Complex[0.08485061223426596, 0.11980477813807357`], {7, 10} -> + Complex[-0.17479296155780177`, 0.14732776197358502`], {7, 11} -> + Complex[0.09710709374348331, 0.20541302638200343`], {7, 12} -> + Complex[-0.04249050665712762, -0.090268967382963], {7, 13} -> + Complex[0.09710709374348331, 0.20541302638200343`], {7, 14} -> + Complex[-0.04249050665712762, -0.090268967382963], {7, 15} -> + Complex[0.08485061223426596, 0.11980477813807357`], {7, 16} -> + Complex[-0.17479296155780177`, 0.14732776197358502`], {7, 17} -> + 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0.161016937967343, -0.024863370726720582`], {1, 13} -> + Complex[-0.16585075463443222`, -0.4085141858532183], {1, 14} -> Complex[ - 0.5691291778553188, -0.009546455336403484], {3, 15} -> - Complex[-0.06237372995183321, -0.06727056435677253], {3, + 0.161016937967343, -0.024863370726720582`], {1, 15} -> + Complex[0.34087862056969404`, -0.1100388283919505], {1, 16} -> Complex[ - 0.3146513247958228, 0.026797378998685423`], {3, 17} -> - Complex[-0.07115780937991523, 0.011205895451017422`], {3, - 18} -> Complex[ - 0.6164020052722983, -0.03301684548082712], {3, 19} -> - Complex[-0.32198385415100245`, 0.056214448752933965`], {3, - 20} -> - Complex[-0.020878723063805726`, -0.1044611112500656], {3, - 21} -> Complex[-0.1157253008903192, \ --0.07182611485724257], {3, 22} -> Complex[0.7322837744378387, 0.], {3, 23} -> - Complex[-0.29340373279675197`, -0.27809938962030384`], {3, - 24} -> Complex[-0.07968743240634073, - 0.17702649633258472`], {4, 5} -> - Complex[0.02087872306380592, -0.10446111125006569`], {4, + 0.33333532926317844`, -0.0052698415932301845`], {1, 17} -> + Complex[0.29687889692943875`, -0.2375326567068352], {1, + 18} -> Complex[-0.42262163352679005`, \ +-0.31456841745062747`], {1, 19} -> + Complex[-0.1599463611986579, -0.3186964622408383], {1, + 20} -> Complex[ + 0.46472596940822686`, -0.06199143840307167], {1, 21} -> + Complex[-0.1766768302665536, 0.1352957631030186], {1, 22} -> + Complex[-0.41976082253431385`, -0.08596477743313696], {1, + 24} -> 1, {2, 3} -> + Complex[-0.06947320790243908, -0.12537107773431694`], {2, + 4} -> Complex[-0.2782895722096578, 0.5061420791848172], { + 2, 5} -> + Complex[-0.06797212261961828, -0.14596193524618625`], {2, 6} -> Complex[ - 0.24014852879671952`, 0.15329539227786965`], {4, 7} -> - Complex[0.07416506719537758, -0.08051189278847616], {4, - 8} -> Complex[-0.2385656438458264, 0.06903700176549488], { - 4, 9} -> Complex[-0.030346695869449408`, - 0.030149558835644692`], {4, 10} -> - Complex[0.011359633728347016`, 0.08520800008182422], {4, - 11} -> Complex[ - 0.09059534447844704, -0.10345647613127328`], {4, 12} -> - Complex[-0.11604982216622163`, 0.19926182212673743`], {4, - 13} -> Complex[ - 0.09059534447844704, -0.10345647613127328`], {4, 14} -> - Complex[-0.11604982216622163`, 0.19926182212673743`], {4, - 15} -> Complex[-0.030346695869449408`, - 0.030149558835644692`], {4, 16} -> - Complex[0.011359633728347016`, 0.08520800008182422], {4, + 0.029019428968993538`, -0.21528502386671927`], {2, 7} -> + Complex[-0.23097227051538433`, 0.42320695740719316`], {2, + 8} -> Complex[-0.16418162796222252`, \ +-0.13133580222113328`], {2, 9} -> + Complex[0.01747597846754094, -0.10203270358947614`], {2, + 10} -> Complex[-0.2917595352077467, -0.3784536838475334], \ +{2, 11} -> Complex[-0.0868392972934217, -0.014513008624828502`], {2, 12} -> + Complex[0.11666092373220538`, 0.218936695047531], {2, 13} -> + Complex[-0.0868392972934217, -0.014513008624828502`], {2, + 14} -> Complex[0.11666092373220538`, 0.218936695047531], { + 2, 15} -> Complex[ + 0.01747597846754094, -0.10203270358947614`], {2, 16} -> + Complex[-0.2917595352077467, -0.3784536838475334], {2, + 17} -> Complex[-0.23097227051538433`, + 0.42320695740719316`], {2, 18} -> + Complex[-0.16418162796222252`, -0.13133580222113328`], {2, + 19} -> + Complex[-0.06797212261961828, -0.14596193524618625`], {2, + 20} -> Complex[ + 0.029019428968993538`, -0.21528502386671927`], {2, 21} -> + Complex[-0.06947320790243908, -0.12537107773431694`], {2, + 22} -> Complex[-0.2782895722096578, 0.5061420791848172], { + 2, 23} -> 1, {3, 4} -> + Complex[0.8762699833354937, 0.], {3, 5} -> + Complex[0.030636763457055183`, 0.11355874224143539`], {3, + 6} -> Complex[-0.5748910514001031, + 0.14633045383134305`], {3, 7} -> + Complex[0.037739555243020126`, 0.012308050579545225`], {3, + 8} -> Complex[ + 0.2616855917631509, 0.009667937264448562], {3, 9} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {3, + 10} -> Complex[-0.2346066684020043, + 0.5335785389713812], {3, 11} -> + Complex[0.25833102475550235`, 0.37839808713437206`], {3, + 12} -> Complex[-0.06250476454855013, \ +-0.34969666557353624`], {3, 13} -> + Complex[0.25833102475550235`, 0.37839808713437206`], {3, + 14} -> Complex[-0.06250476454855013, \ +-0.34969666557353624`], {3, 15} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {3, + 16} -> Complex[-0.2346066684020043, + 0.5335785389713812], {3, 17} -> + Complex[0.037739555243020126`, 0.012308050579545225`], {3, + 18} -> Complex[ + 0.2616855917631509, 0.009667937264448562], {3, 19} -> + Complex[0.030636763457055183`, 0.11355874224143539`], {3, + 20} -> Complex[-0.5748910514001031, + 0.14633045383134305`], {3, 21} -> + Complex[-1.1102230246251565`*^-16, 0.], {3, 22} -> + Complex[0.8762699833354937, 0.], {3, 23} -> + Complex[-0.2782895722096578, -0.5061420791848172], {3, + 24} -> Complex[-0.41976082253431385`, + 0.08596477743313696], {4, 5} -> + Complex[-0.029720518023860276`, 0.053544088868121745`], { + 4, 6} -> Complex[-0.15211930857597808`, + 0.0462428201376506], {4, 7} -> + Complex[-0.13167457410588743`, 0.003054508743256395], {4, + 8} -> Complex[ + 0.053161902759652525`, 0.14451247653181823`], {4, 9} -> + Complex[-0.04368290380765367, 0.027436459786564067`], {4, + 10} -> Complex[ + 0.08210404404737587, -0.05307307167276523], {4, 11} -> + Complex[0.01476498538421066, 0.0004105289599095289], {4, + 12} -> Complex[-0.008146862343863803, \ +-0.10273781027166022`], {4, 13} -> + Complex[0.01476498538421066, 0.0004105289599095289], {4, + 14} -> Complex[-0.008146862343863803, \ +-0.10273781027166022`], {4, 15} -> + Complex[-0.04368290380765367, 0.027436459786564067`], {4, + 16} -> Complex[ + 0.08210404404737587, -0.05307307167276523], {4, 17} -> + Complex[-0.13167457410588743`, 0.003054508743256395], {4, + 18} -> Complex[ + 0.053161902759652525`, 0.14451247653181823`], {4, 19} -> + Complex[-0.029720518023860276`, 0.053544088868121745`], { + 4, 20} -> Complex[-0.15211930857597808`, + 0.0462428201376506], {4, 21} -> + Complex[0.12373001666450631`, 0.], {4, 22} -> + Complex[-1.1102230246251565`*^-16, 0.], {4, 23} -> + Complex[-0.06947320790243908, 0.12537107773431694`], {4, + 24} -> Complex[-0.1766768302665536, -0.1352957631030186], \ +{5, 6} -> Complex[0.7835951260215941, 0.], {5, 7} -> + Complex[0.2578656808252187, -0.10876030801336257`], {5, + 8} -> Complex[-0.17834040611966948`, \ +-0.12609948481270494`], {5, 9} -> + Complex[0.13476314496413244`, 0.0767423516894553], {5, + 10} -> Complex[ + 0.12289396168675226`, -0.38209796458606343`], {5, 11} -> + Complex[-0.3239368353423375, -0.27853835234466173`], {5, + 12} -> Complex[-0.19299966150123138`, + 0.3282026155242766], {5, 13} -> + Complex[-0.3239368353423375, -0.27853835234466173`], {5, + 14} -> Complex[-0.19299966150123138`, + 0.3282026155242766], {5, 15} -> + Complex[0.13476314496413244`, 0.0767423516894553], {5, + 16} -> Complex[ + 0.12289396168675226`, -0.38209796458606343`], {5, 17} -> + Complex[0.2578656808252187, -0.10876030801336257`], {5, + 18} -> Complex[-0.17834040611966948`, \ +-0.12609948481270494`], {5, 19} -> + Complex[ + 8.326672684688674*^-17, -1.3877787807814457`*^-17], {5, + 20} -> Complex[0.7835951260215941, 0.], {5, 21} -> + Complex[-0.15211930857597808`, -0.0462428201376506], {5, + 22} -> Complex[-0.5748910514001031, \ +-0.14633045383134305`], {5, 23} -> + Complex[0.029019428968993538`, 0.21528502386671927`], {5, + 24} -> Complex[ + 0.46472596940822686`, 0.06199143840307167], {6, 7} -> + Complex[-0.007671097858546656, -0.0057340892009139326`], { + 6, 8} -> Complex[ + 0.22516518597054483`, -0.1013577129442031], {6, 9} -> + Complex[0.02081012384838236, 0.14234469229568147`], {6, + 10} -> Complex[-0.09460977915945387, + 0.11933166053315665`], {6, 11} -> + Complex[0.13688251203715338`, 0.1036078618111689], {6, + 12} -> Complex[ + 0.017368709836144688`, 0.1497980533645717], {6, 13} -> + Complex[0.13688251203715338`, 0.1036078618111689], {6, + 14} -> Complex[ + 0.017368709836144688`, 0.1497980533645717], {6, 15} -> + Complex[0.02081012384838236, 0.14234469229568147`], {6, + 16} -> Complex[-0.09460977915945387, + 0.11933166053315665`], {6, 17} -> + Complex[-0.007671097858546656, -0.0057340892009139326`], { + 6, 18} -> Complex[ + 0.22516518597054483`, -0.1013577129442031], {6, 19} -> + Complex[0.21640487397840627`, 0.], {6, 20} -> + Complex[8.326672684688674*^-17, + 1.3877787807814457`*^-17], {6, 21} -> + Complex[-0.029720518023860276`, -0.053544088868121745`], { + 6, 22} -> Complex[ + 0.030636763457055183`, -0.11355874224143539`], {6, 23} -> + Complex[-0.06797212261961828, 0.14596193524618625`], {6, + 24} -> Complex[-0.1599463611986579, 0.3186964622408383], { + 7, 8} -> Complex[0.47738623440613004`, 0.], {7, 9} -> + Complex[0.08485061223426596, 0.11980477813807357`], {7, + 10} -> Complex[-0.17479296155780177`, + 0.14732776197358502`], {7, 11} -> + Complex[0.09710709374348331, 0.20541302638200343`], {7, + 12} -> Complex[-0.04249050665712762, -0.090268967382963], \ +{7, 13} -> Complex[0.09710709374348331, 0.20541302638200343`], {7, 14} -> + Complex[-0.04249050665712762, -0.090268967382963], {7, + 15} -> Complex[ + 0.08485061223426596, 0.11980477813807357`], {7, 16} -> + Complex[-0.17479296155780177`, 0.14732776197358502`], {7, + 17} -> Complex[-1.7867651802561116`*^-16, \ +-1.1449174941446927`*^-16], {7, 18} -> + Complex[0.47738623440613004`, 0.], {7, 19} -> + Complex[0.22516518597054483`, 0.1013577129442031], {7, + 20} -> Complex[-0.17834040611966948`, + 0.12609948481270494`], {7, 21} -> + Complex[0.053161902759652525`, -0.14451247653181823`], {7, + 22} -> Complex[ + 0.2616855917631509, -0.009667937264448562], {7, 23} -> + Complex[-0.16418162796222252`, 0.13133580222113328`], {7, + 24} -> Complex[-0.42262163352679005`, + 0.31456841745062747`], {8, 9} -> + Complex[0.17479296155780188`, 0.14732776197358494`], {8, + 10} -> Complex[-0.08485061223426603, + 0.11980477813807351`], {8, 11} -> + Complex[-0.13183382282185513`, -0.15967490110950291`], {8, + 12} -> Complex[ + 0.11065480592987563`, -0.022498803837048064`], {8, 13} -> + Complex[-0.13183382282185513`, -0.15967490110950291`], {8, + 14} -> Complex[ + 0.11065480592987563`, -0.022498803837048064`], {8, 15} -> + Complex[0.17479296155780188`, 0.14732776197358494`], {8, + 16} -> Complex[-0.08485061223426603, + 0.11980477813807351`], {8, 17} -> + Complex[0.5226137655938699, 0.], {8, 18} -> + Complex[-1.7867651802561116`*^-16, + 1.1449174941446927`*^-16], {8, 19} -> + Complex[-0.007671097858546656, 0.0057340892009139326`], { + 8, 20} -> Complex[ + 0.2578656808252187, 0.10876030801336257`], {8, 21} -> + Complex[-0.13167457410588743`, -0.003054508743256395], {8, + 22} -> Complex[ + 0.037739555243020126`, -0.012308050579545225`], {8, 23} -> + Complex[-0.23097227051538433`, -0.42320695740719316`], {8, + 24} -> Complex[ + 0.29687889692943875`, 0.2375326567068352], {9, 10} -> + Complex[0.619974902703056, 0.], {9, 11} -> + Complex[0.13800495411149877`, -0.3680867713233184], {9, + 12} -> Complex[-0.1905043599805688, \ +-0.13762822610920328`], {9, 13} -> + Complex[0.13800495411149877`, -0.3680867713233184], {9, + 14} -> Complex[-0.1905043599805688, \ +-0.13762822610920328`], {9, 15} -> + Complex[-8.673617379884035*^-17, + 8.326672684688674*^-17], {9, 16} -> + Complex[0.619974902703056, 0.], {9, 17} -> + Complex[-0.08485061223426603, -0.11980477813807351`], {9, + 18} -> Complex[-0.17479296155780177`, \ +-0.14732776197358502`], {9, 19} -> + Complex[-0.09460977915945387, -0.11933166053315665`], {9, + 20} -> Complex[ + 0.12289396168675226`, 0.38209796458606343`], {9, 21} -> + Complex[0.08210404404737587, 0.05307307167276523], {9, + 22} -> Complex[-0.2346066684020043, -0.5335785389713812], \ +{9, 23} -> Complex[-0.2917595352077467, 0.3784536838475334], {9, 24} -> + Complex[0.33333532926317844`, 0.0052698415932301845`], { + 10, 11} -> + Complex[-0.15558917647409173`, -0.09979098803090053], {10, + 12} -> Complex[ + 0.1899783589314758, -0.1365082746823964], {10, 13} -> + Complex[-0.15558917647409173`, -0.09979098803090053], {10, + 14} -> Complex[ + 0.1899783589314758, -0.1365082746823964], {10, 15} -> + Complex[0.3800250972969439, 0.], {10, 16} -> + Complex[-8.673617379884035*^-17, \ +-8.326672684688674*^-17], {10, 17} -> + Complex[0.17479296155780188`, -0.14732776197358494`], {10, + 18} -> Complex[ + 0.08485061223426596, -0.11980477813807357`], {10, 19} -> + Complex[0.02081012384838236, -0.14234469229568147`], {10, + 20} -> Complex[ + 0.13476314496413244`, -0.0767423516894553], {10, 21} -> + Complex[-0.04368290380765367, -0.027436459786564067`], { + 10, 22} -> + Complex[-0.15157725194981503`, 0.17844414940708186`], {10, + 23} -> Complex[ + 0.01747597846754094, 0.10203270358947614`], {10, 24} -> + Complex[0.34087862056969404`, 0.1100388283919505], {11, + 12} -> Complex[0.5275300911030236, 0.], {11, 13} -> + Complex[1.8041124150158796`*^-16, + 1.457167719820518*^-16], {11, 14} -> + Complex[0.5275300911030236, 0.], {11, 15} -> + Complex[0.1899783589314758, 0.1365082746823964], {11, 16} -> + Complex[-0.1905043599805688, 0.13762822610920328`], {11, 17} -> Complex[ - 0.07416506719537758, -0.08051189278847616], {4, 18} -> - Complex[-0.2385656438458264, 0.06903700176549488], {4, + 0.11065480592987563`, 0.022498803837048064`], {11, 18} -> + Complex[-0.04249050665712762, 0.090268967382963], {11, 19} -> Complex[ - 0.02087872306380592, -0.10446111125006569`], {4, 20} -> - Complex[0.24014852879671952`, 0.15329539227786965`], {4, - 21} -> Complex[0.2677162255621613, 0.], {4, 22} -> - Complex[-0.1157253008903192, 0.07182611485724257], {4, + 0.017368709836144688`, -0.1497980533645717], {11, 20} -> + Complex[-0.19299966150123138`, -0.3282026155242766], {11, + 21} -> Complex[-0.008146862343863803, + 0.10273781027166022`], {11, 22} -> + Complex[-0.06250476454855013, 0.34969666557353624`], {11, 23} -> Complex[ - 0.27244326195988916`, -0.011111878577649018`], {4, 24} -> - Complex[-0.3506829757791454, -0.10726001244941838`], {5, - 6} -> Complex[0.460087813322552, 0.], {5, 7} -> - Complex[-0.03764992141799037, -0.13605099778794436`], {5, - 8} -> Complex[-0.02707098756043888, - 0.08415309369657159], {5, 9} -> - Complex[0.22168267144312054`, 0.03299920735716682], {5, - 10} -> Complex[-0.08181885113853848, \ --0.0223150746441549], {5, 11} -> - Complex[0.14142597329959572`, -0.061350391806736695`], {5, - 12} -> Complex[-0.04214475287108173, - 0.2125642839376449], {5, 13} -> - Complex[0.14142597329959572`, -0.061350391806736695`], {5, - 14} -> Complex[-0.04214475287108173, - 0.2125642839376449], {5, 15} -> - Complex[0.22168267144312054`, 0.03299920735716682], {5, - 16} -> Complex[-0.08181885113853848, \ --0.0223150746441549], {5, 17} -> - Complex[-0.03764992141799037, -0.13605099778794436`], {5, - 18} -> Complex[-0.02707098756043888, - 0.08415309369657159], {5, 19} -> - Complex[-0.27467400107696954`, -0.08314412135251599], {5, - 20} -> Complex[0.460087813322552, 0.], {5, 21} -> - Complex[0.24014852879671952`, -0.15329539227786965`], {5, - 22} -> Complex[-0.020878723063805726`, - 0.1044611112500656], {5, 23} -> - Complex[0.11519091687891489`, -0.4012928958026726], {5, - 24} -> Complex[-0.45588071484204273`, - 0.014752170920114424`], {6, 7} -> - Complex[0.12702736104247997`, 0.01646149694625006], {6, - 8} -> Complex[-0.4024303768625533, 0.06319936655428628], { - 6, 9} -> Complex[-0.2934854494126851, - 0.04085502210736608], {6, 10} -> - Complex[0.0806509155236052, 0.06238441832735686], {6, 11} -> - Complex[-0.029186234486845222`, -0.022372160811117286`], { - 6, 12} -> - Complex[-0.18425433538775948`, -0.03665899040288985], {6, - 13} -> Complex[-0.029186234486845222`, \ --0.022372160811117286`], {6, 14} -> - Complex[-0.18425433538775948`, -0.03665899040288985], {6, - 15} -> Complex[-0.2934854494126851, - 0.04085502210736608], {6, 16} -> - Complex[0.0806509155236052, 0.06238441832735686], {6, 17} -> - Complex[0.12702736104247997`, 0.01646149694625006], {6, - 18} -> Complex[-0.4024303768625533, - 0.06319936655428628], {6, 19} -> - Complex[0.5399121866774477, 0.], {6, 20} -> - Complex[-0.27467400107696954`, 0.08314412135251599], {6, + 0.11666092373220538`, -0.218936695047531], {11, 24} -> + Complex[0.161016937967343, 0.024863370726720582`], {12, + 13} -> Complex[0.4724699088969763, 0.], {12, 14} -> + Complex[ + 1.8041124150158796`*^-16, -1.457167719820518*^-16], {12, + 15} -> Complex[-0.15558917647409173`, + 0.09979098803090053], {12, 16} -> + Complex[0.13800495411149877`, 0.3680867713233184], {12, + 17} -> Complex[-0.13183382282185513`, + 0.15967490110950291`], {12, 18} -> + Complex[0.09710709374348331, -0.20541302638200343`], {12, + 19} -> Complex[ + 0.13688251203715338`, -0.1036078618111689], {12, 20} -> + Complex[-0.3239368353423375, 0.27853835234466173`], {12, 21} -> Complex[ - 0.02087872306380592, 0.10446111125006569`], {6, 22} -> - Complex[-0.32198385415100245`, 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Complex[-0.2385656438458264, -0.06903700176549488], {7, - 22} -> Complex[0.6164020052722983, 0.03301684548082712], { - 7, 23} -> - Complex[-0.24737746704016716`, -0.5022802985841684], {7, - 24} -> Complex[0.199408104684111, 0.11720650121625686`], { - 8, 9} -> Complex[-0.11463533571593523`, - 0.055210078698593804`], {8, 10} -> - Complex[0.06911881436797135, 0.04269376157510484], {8, - 11} -> Complex[-0.010533198723196371`, \ --0.006296053518233197], {8, 12} -> - Complex[-0.06875971685231415, 0.006550328119388341], {8, - 13} -> Complex[-0.010533198723196371`, \ --0.006296053518233197], {8, 14} -> - Complex[-0.06875971685231415, 0.006550328119388341], {8, - 15} -> Complex[-0.11463533571593523`, - 0.055210078698593804`], {8, 16} -> - Complex[0.06911881436797135, 0.04269376157510484], {8, - 17} -> Complex[0.09630860197729411, 0.], {8, 18} -> - Complex[-0.13812244808851923`, 0.022678210677030422`], {8, - 19} -> Complex[ - 0.12702736104247997`, -0.01646149694625006], {8, 20} -> - Complex[-0.03764992141799037, 0.13605099778794436`], {8, + 0.01476498538421066, -0.0004105289599095289], {12, 22} -> + Complex[0.25833102475550235`, -0.37839808713437206`], {12, + 23} -> Complex[-0.0868392972934217, + 0.014513008624828502`], {12, 24} -> + Complex[-0.16585075463443222`, 0.4085141858532183], {13, + 14} -> Complex[0.5275300911030236, 0.], {13, 15} -> + Complex[0.1899783589314758, 0.1365082746823964], {13, 16} -> + Complex[-0.1905043599805688, 0.13762822610920328`], {13, + 17} -> Complex[ + 0.11065480592987563`, 0.022498803837048064`], {13, 18} -> + Complex[-0.04249050665712762, 0.090268967382963], {13, + 19} -> Complex[ + 0.017368709836144688`, -0.1497980533645717], {13, 20} -> + Complex[-0.19299966150123138`, -0.3282026155242766], {13, + 21} -> Complex[-0.008146862343863803, + 0.10273781027166022`], {13, 22} -> + Complex[-0.06250476454855013, 0.34969666557353624`], {13, + 23} -> Complex[ + 0.11666092373220538`, -0.218936695047531], {13, 24} -> + Complex[0.161016937967343, 0.024863370726720582`], {14, + 15} -> Complex[-0.15558917647409173`, + 0.09979098803090053], {14, 16} -> + Complex[0.13800495411149877`, 0.3680867713233184], {14, + 17} -> Complex[-0.13183382282185513`, + 0.15967490110950291`], {14, 18} -> + Complex[0.09710709374348331, -0.20541302638200343`], {14, + 19} -> Complex[ + 0.13688251203715338`, -0.1036078618111689], {14, 20} -> + Complex[-0.3239368353423375, 0.27853835234466173`], {14, 21} -> Complex[ - 0.07416506719537758, 0.08051189278847616], {8, 22} -> - Complex[-0.07115780937991523, -0.011205895451017422`], {8, + 0.01476498538421066, -0.0004105289599095289], {14, 22} -> + Complex[0.25833102475550235`, -0.37839808713437206`], {14, + 23} -> Complex[-0.0868392972934217, + 0.014513008624828502`], {14, 24} -> + Complex[-0.16585075463443222`, 0.4085141858532183], {15, + 16} -> Complex[0.619974902703056, 0.], {15, 17} -> + Complex[-0.08485061223426603, -0.11980477813807351`], {15, + 18} -> + Complex[-0.17479296155780177`, -0.14732776197358502`], { + 15, 19} -> + Complex[-0.09460977915945387, -0.11933166053315665`], {15, + 20} -> Complex[ + 0.12289396168675226`, 0.38209796458606343`], {15, 21} -> + Complex[0.08210404404737587, 0.05307307167276523], {15, + 22} -> Complex[-0.2346066684020043, -0.5335785389713812], \ +{15, 23} -> Complex[-0.2917595352077467, 0.3784536838475334], {15, 24} -> + Complex[0.33333532926317844`, 0.0052698415932301845`], { + 16, 17} -> + Complex[0.17479296155780188`, -0.14732776197358494`], {16, + 18} -> Complex[ + 0.08485061223426596, -0.11980477813807357`], {16, 19} -> + Complex[0.02081012384838236, -0.14234469229568147`], {16, + 20} -> Complex[ + 0.13476314496413244`, -0.0767423516894553], {16, 21} -> + Complex[-0.04368290380765367, -0.027436459786564067`], { + 16, 22} -> + Complex[-0.15157725194981503`, 0.17844414940708186`], {16, 23} -> Complex[ - 0.16774613988004144`, 0.225655277072902], {8, 24} -> - Complex[-0.09714752480970831, -0.05391368675976281], {9, - 10} -> Complex[0.4159087185308232, 0.], {9, 11} -> - Complex[-0.364332505630415, -0.055629112414147225`], {9, - 12} -> Complex[ - 0.3689066714841379, -0.01963818646965646], {9, 13} -> - Complex[-0.364332505630415, -0.055629112414147225`], {9, - 14} -> Complex[ - 0.3689066714841379, -0.01963818646965646], {9, 15} -> - Complex[-0.29089675324323006`, 0.1558046632134409], {9, - 16} -> Complex[0.4159087185308232, 0.], {9, 17} -> - Complex[0.06911881436797135, -0.04269376157510484], {9, - 18} -> Complex[ - 0.14880448551485936`, 0.18343023977218464`], {9, 19} -> - Complex[0.0806509155236052, -0.06238441832735686], {9, - 20} -> Complex[-0.08181885113853848, - 0.0223150746441549], {9, 21} -> - Complex[0.011359633728347016`, -0.08520800008182422], {9, - 22} -> Complex[ - 0.3146513247958228, -0.026797378998685423`], {9, 23} -> - Complex[0.17837001887568327`, 0.09187489939370964], {9, - 24} -> Complex[-0.07164664525064873, - 0.2907134789193454], {10, 11} -> - Complex[0.24390216997712094`, 0.12967162455970677`], {10, - 12} -> Complex[-0.1634450723422762, \ --0.0011076658174966336`], {10, 13} -> - Complex[0.24390216997712094`, 0.12967162455970677`], {10, - 14} -> Complex[-0.1634450723422762, \ --0.0011076658174966336`], {10, 15} -> - Complex[0.41416215192864014`, 0.], {10, 16} -> - Complex[-0.29089675324323006`, -0.1558046632134409], {10, - 17} -> Complex[-0.11463533571593523`, \ --0.055210078698593804`], {10, 18} -> - Complex[0.1868664022204954, -0.06385028540854319], {10, - 19} -> Complex[-0.2934854494126851, \ --0.04085502210736608], {10, 20} -> - Complex[0.22168267144312054`, -0.03299920735716682], {10, - 21} -> Complex[-0.030346695869449408`, \ --0.030149558835644692`], {10, 22} -> - Complex[-0.06237372995183321, 0.06727056435677253], {10, - 23} -> Complex[-0.09727281618814543, \ --0.4251726947214757], {10, 24} -> - Complex[0.02023138254715448, -0.11137525641609], {11, 12} -> - Complex[0.5794387931418044, 0.], {11, 13} -> - Complex[-0.4009452410407828, -0.11284019268466222`], {11, - 14} -> Complex[0.5794387931418044, 0.], {11, 15} -> - Complex[-0.1634450723422762, 0.0011076658174966336`], {11, - 16} -> Complex[ - 0.3689066714841379, 0.01963818646965646], {11, 17} -> - Complex[-0.06875971685231415, -0.006550328119388341], {11, - 18} -> Complex[0.4351121808870284, 0.1282880779202612], { - 11, 19} -> - Complex[-0.18425433538775948`, 0.03665899040288985], {11, - 20} -> Complex[-0.04214475287108173, \ --0.2125642839376449], {11, 21} -> - Complex[-0.11604982216622163`, -0.19926182212673743`], { - 11, 22} -> - Complex[0.5691291778553188, 0.009546455336403484], {11, - 23} -> Complex[-0.20476687273928368`, \ --0.20940927658482447`], {11, 24} -> - Complex[-0.04633762187626639, 0.37156097014999256`], {12, - 13} -> Complex[0.3744935743083003, 0.], {12, 14} -> - Complex[-0.4009452410407828, 0.11284019268466222`], {12, - 15} -> Complex[ - 0.24390216997712094`, -0.12967162455970677`], {12, 16} -> - Complex[-0.364332505630415, 0.055629112414147225`], {12, - 17} -> Complex[-0.010533198723196371`, - 0.006296053518233197], {12, 18} -> - Complex[-0.2300764357164763, -0.11137567802408056`], {12, - 19} -> Complex[-0.029186234486845222`, - 0.022372160811117286`], {12, 20} -> - Complex[0.14142597329959572`, 0.061350391806736695`], {12, - 21} -> Complex[ - 0.09059534447844704, 0.10345647613127328`], {12, 22} -> - Complex[-0.33684082460892883`, 0.09005315780013691], {12, - 23} -> Complex[-0.008851768247553363, \ --0.029068202106085572`], {12, 24} -> - Complex[-0.07795077274116374, -0.2904671042221048], {13, - 14} -> Complex[0.5794387931418044, 0.], {13, 15} -> - Complex[-0.1634450723422762, 0.0011076658174966336`], {13, - 16} -> Complex[ - 0.3689066714841379, 0.01963818646965646], {13, 17} -> - Complex[-0.06875971685231415, -0.006550328119388341], {13, - 18} -> Complex[0.4351121808870284, 0.1282880779202612], { - 13, 19} -> - Complex[-0.18425433538775948`, 0.03665899040288985], {13, - 20} -> Complex[-0.04214475287108173, \ --0.2125642839376449], {13, 21} -> - Complex[-0.11604982216622163`, -0.19926182212673743`], { - 13, 22} -> - Complex[0.5691291778553188, 0.009546455336403484], {13, - 23} -> Complex[-0.20476687273928368`, \ --0.20940927658482447`], {13, 24} -> - Complex[-0.04633762187626639, 0.37156097014999256`], {14, - 15} -> Complex[ - 0.24390216997712094`, -0.12967162455970677`], {14, 16} -> - Complex[-0.364332505630415, 0.055629112414147225`], {14, - 17} -> Complex[-0.010533198723196371`, - 0.006296053518233197], {14, 18} -> - Complex[-0.2300764357164763, -0.11137567802408056`], {14, - 19} -> Complex[-0.029186234486845222`, - 0.022372160811117286`], {14, 20} -> - Complex[0.14142597329959572`, 0.061350391806736695`], {14, - 21} -> Complex[ - 0.09059534447844704, 0.10345647613127328`], {14, 22} -> - Complex[-0.33684082460892883`, 0.09005315780013691], {14, - 23} -> Complex[-0.008851768247553363, \ --0.029068202106085572`], {14, 24} -> - Complex[-0.07795077274116374, -0.2904671042221048], {15, - 16} -> Complex[0.4159087185308232, 0.], {15, 17} -> - Complex[0.06911881436797135, -0.04269376157510484], {15, - 18} -> Complex[ - 0.14880448551485936`, 0.18343023977218464`], {15, 19} -> - Complex[0.0806509155236052, -0.06238441832735686], {15, - 20} -> Complex[-0.08181885113853848, - 0.0223150746441549], {15, 21} -> - Complex[0.011359633728347016`, -0.08520800008182422], {15, - 22} -> Complex[ - 0.3146513247958228, -0.026797378998685423`], {15, 23} -> - Complex[0.17837001887568327`, 0.09187489939370964], {15, - 24} -> Complex[-0.07164664525064873, - 0.2907134789193454], {16, 17} -> - Complex[-0.11463533571593523`, -0.055210078698593804`], { - 16, 18} -> - Complex[0.1868664022204954, -0.06385028540854319], {16, - 19} -> Complex[-0.2934854494126851, \ --0.04085502210736608], {16, 20} -> - Complex[0.22168267144312054`, -0.03299920735716682], {16, - 21} -> Complex[-0.030346695869449408`, \ --0.030149558835644692`], {16, 22} -> - Complex[-0.06237372995183321, 0.06727056435677253], {16, - 23} -> Complex[-0.09727281618814543, \ --0.4251726947214757], {16, 24} -> - Complex[0.02023138254715448, -0.11137525641609], {17, 18} -> - Complex[0.7694387170582185, 0.], {17, 19} -> - Complex[-0.4024303768625533, -0.06319936655428628], {17, - 20} -> Complex[-0.02707098756043888, \ --0.08415309369657159], {17, 21} -> - Complex[-0.2385656438458264, -0.06903700176549488], {17, - 22} -> Complex[0.6164020052722983, 0.03301684548082712], { - 17, 23} -> - Complex[-0.24737746704016716`, -0.5022802985841684], {17, - 24} -> Complex[0.199408104684111, 0.11720650121625686`], { - 18, 19} -> - Complex[0.12702736104247997`, -0.01646149694625006], {18, - 20} -> Complex[-0.03764992141799037, - 0.13605099778794436`], {18, 21} -> - Complex[0.07416506719537758, 0.08051189278847616], {18, - 22} -> Complex[-0.07115780937991523, \ --0.011205895451017422`], {18, 23} -> - Complex[0.16774613988004144`, 0.225655277072902], {18, - 24} -> Complex[-0.09714752480970831, \ --0.05391368675976281], {19, 20} -> Complex[0.460087813322552, 0.], {19, 21} -> - Complex[0.24014852879671952`, -0.15329539227786965`], {19, - 22} -> Complex[-0.020878723063805726`, - 0.1044611112500656], {19, 23} -> - Complex[0.11519091687891489`, -0.4012928958026726], {19, - 24} -> Complex[-0.45588071484204273`, - 0.014752170920114424`], {20, 21} -> - Complex[0.02087872306380592, 0.10446111125006569`], {20, - 22} -> Complex[-0.32198385415100245`, \ --0.056214448752933965`], {20, 23} -> - Complex[0.34202714802636813`, 0.5203828231210746], {20, - 24} -> Complex[ - 0.27130379014783124`, 0.12044715596755934`], {21, 22} -> - Complex[0.7322837744378387, 0.], {21, 23} -> - Complex[-0.29340373279675197`, -0.27809938962030384`], { - 21, 24} -> - Complex[-0.07968743240634073, 0.17702649633258472`], {22, - 23} -> Complex[ - 0.27244326195988916`, -0.011111878577649018`], {22, 24} -> - Complex[-0.3506829757791454, -0.10726001244941838`]}, + 0.01747597846754094, 0.10203270358947614`], {16, 24} -> + Complex[0.34087862056969404`, 0.1100388283919505], {17, + 18} -> Complex[0.47738623440613004`, 0.], {17, 19} -> + Complex[0.22516518597054483`, 0.1013577129442031], {17, + 20} -> Complex[-0.17834040611966948`, + 0.12609948481270494`], {17, 21} -> + Complex[0.053161902759652525`, -0.14451247653181823`], { + 17, 22} -> + Complex[0.2616855917631509, -0.009667937264448562], {17, + 23} -> Complex[-0.16418162796222252`, + 0.13133580222113328`], {17, 24} -> + Complex[-0.42262163352679005`, 0.31456841745062747`], {18, + 19} -> Complex[-0.007671097858546656, + 0.0057340892009139326`], {18, 20} -> + Complex[0.2578656808252187, 0.10876030801336257`], {18, + 21} -> Complex[-0.13167457410588743`, \ +-0.003054508743256395], {18, 22} -> + Complex[0.037739555243020126`, -0.012308050579545225`], { + 18, 23} -> + Complex[-0.23097227051538433`, -0.42320695740719316`], { + 18, 24} -> + Complex[0.29687889692943875`, 0.2375326567068352], {19, + 20} -> 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Complex[0.7694387170582185, 0.], {17, 19} -> - Complex[-0.4024303768625533, -0.06319936655428628], {17, 20} -> - Complex[-0.02707098756043888, -0.08415309369657159], {17, 21} -> - Complex[-0.2385656438458264, -0.06903700176549488], {17, 22} -> - Complex[0.6164020052722983, 0.03301684548082712], {17, 23} -> - Complex[-0.24737746704016716`, -0.5022802985841684], {17, 24} -> - Complex[0.199408104684111, 0.11720650121625686`], {18, 19} -> - Complex[0.12702736104247997`, -0.01646149694625006], {18, 20} -> - Complex[-0.03764992141799037, 0.13605099778794436`], {18, 21} -> - Complex[0.07416506719537758, 0.08051189278847616], {18, 22} -> - Complex[-0.07115780937991523, -0.011205895451017422`], {18, 23} -> - Complex[0.16774613988004144`, 0.225655277072902], {18, 24} -> - Complex[-0.09714752480970831, -0.05391368675976281], {19, 20} -> - Complex[0.460087813322552, 0.], {19, 21} -> - Complex[0.24014852879671952`, -0.15329539227786965`], {19, 22} -> - Complex[-0.020878723063805726`, 0.1044611112500656], {19, 23} -> - Complex[0.11519091687891489`, -0.4012928958026726], {19, 24} -> - Complex[-0.45588071484204273`, 0.014752170920114424`], {20, 21} -> - Complex[0.02087872306380592, 0.10446111125006569`], {20, 22} -> - Complex[-0.32198385415100245`, -0.056214448752933965`], {20, 23} -> - Complex[0.34202714802636813`, 0.5203828231210746], {20, 24} -> - Complex[0.27130379014783124`, 0.12044715596755934`], {21, 22} -> - Complex[0.7322837744378387, 0.], {21, 23} -> - Complex[-0.29340373279675197`, -0.27809938962030384`], {21, 24} -> - Complex[-0.07968743240634073, 0.17702649633258472`], {22, 23} -> - Complex[0.27244326195988916`, -0.011111878577649018`], {22, 24} -> - Complex[-0.3506829757791454, -0.10726001244941838`]}, + Complex[-0.06947320790243908, -0.12537107773431694`], {2, 4} -> + Complex[-0.2782895722096578, 0.5061420791848172], {2, 5} -> + Complex[-0.06797212261961828, -0.14596193524618625`], {2, 6} -> + Complex[0.029019428968993538`, -0.21528502386671927`], {2, 7} -> + Complex[-0.23097227051538433`, 0.42320695740719316`], {2, 8} -> + Complex[-0.16418162796222252`, -0.13133580222113328`], {2, 9} -> + Complex[0.01747597846754094, -0.10203270358947614`], {2, 10} -> + Complex[-0.2917595352077467, -0.3784536838475334], {2, 11} -> + Complex[-0.0868392972934217, -0.014513008624828502`], {2, 12} -> + Complex[0.11666092373220538`, 0.218936695047531], {2, 13} -> + Complex[-0.0868392972934217, -0.014513008624828502`], {2, 14} -> + Complex[0.11666092373220538`, 0.218936695047531], {2, 15} -> + Complex[0.01747597846754094, -0.10203270358947614`], {2, 16} -> + Complex[-0.2917595352077467, -0.3784536838475334], {2, 17} -> + Complex[-0.23097227051538433`, 0.42320695740719316`], {2, 18} -> + Complex[-0.16418162796222252`, -0.13133580222113328`], {2, 19} -> + Complex[-0.06797212261961828, -0.14596193524618625`], {2, 20} -> + Complex[0.029019428968993538`, -0.21528502386671927`], {2, 21} -> + Complex[-0.06947320790243908, -0.12537107773431694`], {2, 22} -> + Complex[-0.2782895722096578, 0.5061420791848172], {2, 23} -> + 1, {3, 4} -> Complex[0.8762699833354937, 0.], {3, 5} -> + Complex[0.030636763457055183`, 0.11355874224143539`], {3, 6} -> + Complex[-0.5748910514001031, 0.14633045383134305`], {3, 7} -> + Complex[0.037739555243020126`, 0.012308050579545225`], {3, 8} -> + Complex[0.2616855917631509, 0.009667937264448562], {3, 9} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {3, 10} -> + Complex[-0.2346066684020043, 0.5335785389713812], {3, 11} -> + Complex[0.25833102475550235`, 0.37839808713437206`], {3, 12} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {3, 13} -> + Complex[0.25833102475550235`, 0.37839808713437206`], {3, 14} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {3, 15} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {3, 16} -> + Complex[-0.2346066684020043, 0.5335785389713812], {3, 17} -> + Complex[0.037739555243020126`, 0.012308050579545225`], {3, 18} -> + Complex[0.2616855917631509, 0.009667937264448562], {3, 19} -> + Complex[0.030636763457055183`, 0.11355874224143539`], {3, 20} -> + Complex[-0.5748910514001031, 0.14633045383134305`], {3, 21} -> + Complex[-1.1102230246251565`*^-16, 0.], {3, 22} -> + Complex[0.8762699833354937, 0.], {3, 23} -> + Complex[-0.2782895722096578, -0.5061420791848172], {3, 24} -> + Complex[-0.41976082253431385`, 0.08596477743313696], {4, 5} -> + Complex[-0.029720518023860276`, 0.053544088868121745`], {4, 6} -> + Complex[-0.15211930857597808`, 0.0462428201376506], {4, 7} -> + Complex[-0.13167457410588743`, 0.003054508743256395], {4, 8} -> + Complex[0.053161902759652525`, 0.14451247653181823`], {4, 9} -> + Complex[-0.04368290380765367, 0.027436459786564067`], {4, 10} -> + Complex[0.08210404404737587, -0.05307307167276523], {4, 11} -> + Complex[0.01476498538421066, 0.0004105289599095289], {4, 12} -> + Complex[-0.008146862343863803, -0.10273781027166022`], {4, 13} -> + Complex[0.01476498538421066, 0.0004105289599095289], {4, 14} -> + Complex[-0.008146862343863803, -0.10273781027166022`], {4, 15} -> + Complex[-0.04368290380765367, 0.027436459786564067`], {4, 16} -> + Complex[0.08210404404737587, -0.05307307167276523], {4, 17} -> + Complex[-0.13167457410588743`, 0.003054508743256395], {4, 18} -> + Complex[0.053161902759652525`, 0.14451247653181823`], {4, 19} -> + Complex[-0.029720518023860276`, 0.053544088868121745`], {4, 20} -> + Complex[-0.15211930857597808`, 0.0462428201376506], {4, 21} -> + Complex[0.12373001666450631`, 0.], {4, 22} -> + Complex[-1.1102230246251565`*^-16, 0.], {4, 23} -> + Complex[-0.06947320790243908, 0.12537107773431694`], {4, 24} -> + Complex[-0.1766768302665536, -0.1352957631030186], {5, 6} -> + Complex[0.7835951260215941, 0.], {5, 7} -> + Complex[0.2578656808252187, -0.10876030801336257`], {5, 8} -> + Complex[-0.17834040611966948`, -0.12609948481270494`], {5, 9} -> + Complex[0.13476314496413244`, 0.0767423516894553], {5, 10} -> + Complex[0.12289396168675226`, -0.38209796458606343`], {5, 11} -> + Complex[-0.3239368353423375, -0.27853835234466173`], {5, 12} -> + Complex[-0.19299966150123138`, 0.3282026155242766], {5, 13} -> + Complex[-0.3239368353423375, -0.27853835234466173`], {5, 14} -> + Complex[-0.19299966150123138`, 0.3282026155242766], {5, 15} -> + Complex[0.13476314496413244`, 0.0767423516894553], {5, 16} -> + Complex[0.12289396168675226`, -0.38209796458606343`], {5, 17} -> + Complex[0.2578656808252187, -0.10876030801336257`], {5, 18} -> + Complex[-0.17834040611966948`, -0.12609948481270494`], {5, 19} -> + Complex[8.326672684688674*^-17, -1.3877787807814457`*^-17], {5, + 20} -> Complex[0.7835951260215941, 0.], {5, 21} -> + Complex[-0.15211930857597808`, -0.0462428201376506], {5, 22} -> + Complex[-0.5748910514001031, -0.14633045383134305`], {5, 23} -> + Complex[0.029019428968993538`, 0.21528502386671927`], {5, 24} -> + Complex[0.46472596940822686`, 0.06199143840307167], {6, 7} -> + Complex[-0.007671097858546656, -0.0057340892009139326`], {6, 8} -> + Complex[0.22516518597054483`, -0.1013577129442031], {6, 9} -> + Complex[0.02081012384838236, 0.14234469229568147`], {6, 10} -> + Complex[-0.09460977915945387, 0.11933166053315665`], {6, 11} -> + Complex[0.13688251203715338`, 0.1036078618111689], {6, 12} -> + Complex[0.017368709836144688`, 0.1497980533645717], {6, 13} -> + Complex[0.13688251203715338`, 0.1036078618111689], {6, 14} -> + Complex[0.017368709836144688`, 0.1497980533645717], {6, 15} -> + Complex[0.02081012384838236, 0.14234469229568147`], {6, 16} -> + Complex[-0.09460977915945387, 0.11933166053315665`], {6, 17} -> + Complex[-0.007671097858546656, -0.0057340892009139326`], {6, 18} -> + Complex[0.22516518597054483`, -0.1013577129442031], {6, 19} -> + Complex[0.21640487397840627`, 0.], {6, 20} -> + Complex[8.326672684688674*^-17, 1.3877787807814457`*^-17], {6, 21} -> + Complex[-0.029720518023860276`, -0.053544088868121745`], {6, 22} -> + Complex[0.030636763457055183`, -0.11355874224143539`], {6, 23} -> + Complex[-0.06797212261961828, 0.14596193524618625`], {6, 24} -> + Complex[-0.1599463611986579, 0.3186964622408383], {7, 8} -> + Complex[0.47738623440613004`, 0.], {7, 9} -> + Complex[0.08485061223426596, 0.11980477813807357`], {7, 10} -> + Complex[-0.17479296155780177`, 0.14732776197358502`], {7, 11} -> + Complex[0.09710709374348331, 0.20541302638200343`], {7, 12} -> + Complex[-0.04249050665712762, -0.090268967382963], {7, 13} -> + Complex[0.09710709374348331, 0.20541302638200343`], {7, 14} -> + Complex[-0.04249050665712762, -0.090268967382963], {7, 15} -> + Complex[0.08485061223426596, 0.11980477813807357`], {7, 16} -> + Complex[-0.17479296155780177`, 0.14732776197358502`], {7, 17} -> + Complex[-1.7867651802561116`*^-16, -1.1449174941446927`*^-16], {7, + 18} -> Complex[0.47738623440613004`, 0.], {7, 19} -> + Complex[0.22516518597054483`, 0.1013577129442031], {7, 20} -> + Complex[-0.17834040611966948`, 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0.0057340892009139326`], {8, + 20} -> Complex[0.2578656808252187, 0.10876030801336257`], {8, + 21} -> Complex[-0.13167457410588743`, -0.003054508743256395], {8, + 22} -> Complex[0.037739555243020126`, -0.012308050579545225`], {8, + 23} -> Complex[-0.23097227051538433`, -0.42320695740719316`], {8, + 24} -> Complex[0.29687889692943875`, 0.2375326567068352], {9, + 10} -> Complex[0.619974902703056, 0.], {9, 11} -> + Complex[0.13800495411149877`, -0.3680867713233184], {9, 12} -> + Complex[-0.1905043599805688, -0.13762822610920328`], {9, 13} -> + Complex[0.13800495411149877`, -0.3680867713233184], {9, 14} -> + Complex[-0.1905043599805688, -0.13762822610920328`], {9, 15} -> + Complex[-8.673617379884035*^-17, 8.326672684688674*^-17], {9, 16} -> + Complex[0.619974902703056, 0.], {9, 17} -> + Complex[-0.08485061223426603, -0.11980477813807351`], {9, 18} -> + Complex[-0.17479296155780177`, -0.14732776197358502`], {9, 19} -> + Complex[-0.09460977915945387, -0.11933166053315665`], {9, 20} -> + Complex[0.12289396168675226`, 0.38209796458606343`], {9, 21} -> + Complex[0.08210404404737587, 0.05307307167276523], {9, 22} -> + Complex[-0.2346066684020043, -0.5335785389713812], {9, 23} -> + Complex[-0.2917595352077467, 0.3784536838475334], {9, 24} -> + Complex[0.33333532926317844`, 0.0052698415932301845`], {10, 11} -> + Complex[-0.15558917647409173`, -0.09979098803090053], {10, 12} -> + Complex[0.1899783589314758, -0.1365082746823964], {10, 13} -> + Complex[-0.15558917647409173`, -0.09979098803090053], {10, 14} -> + Complex[0.1899783589314758, -0.1365082746823964], {10, 15} -> + Complex[0.3800250972969439, 0.], {10, 16} -> + Complex[-8.673617379884035*^-17, -8.326672684688674*^-17], {10, + 17} -> Complex[0.17479296155780188`, -0.14732776197358494`], {10, + 18} -> Complex[0.08485061223426596, -0.11980477813807357`], {10, + 19} -> Complex[0.02081012384838236, -0.14234469229568147`], {10, + 20} -> Complex[0.13476314496413244`, -0.0767423516894553], {10, + 21} -> Complex[-0.04368290380765367, -0.027436459786564067`], {10, + 22} -> Complex[-0.15157725194981503`, 0.17844414940708186`], {10, + 23} -> + Complex[0.01747597846754094, 0.10203270358947614`], {10, 24} -> + Complex[0.34087862056969404`, 0.1100388283919505], {11, 12} -> + Complex[0.5275300911030236, 0.], {11, 13} -> + Complex[1.8041124150158796`*^-16, 1.457167719820518*^-16], {11, + 14} -> Complex[0.5275300911030236, 0.], {11, 15} -> + Complex[0.1899783589314758, 0.1365082746823964], {11, 16} -> + Complex[-0.1905043599805688, 0.13762822610920328`], {11, 17} -> + Complex[0.11065480592987563`, 0.022498803837048064`], {11, 18} -> + Complex[-0.04249050665712762, 0.090268967382963], {11, 19} -> + Complex[0.017368709836144688`, -0.1497980533645717], {11, 20} -> + Complex[-0.19299966150123138`, -0.3282026155242766], {11, 21} -> + Complex[-0.008146862343863803, 0.10273781027166022`], {11, 22} -> + Complex[-0.06250476454855013, 0.34969666557353624`], {11, 23} -> + Complex[0.11666092373220538`, 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TagBox["NambuMatrices", "SummaryHead"], "[", + TagBox["BogoliubovTransforms", "SummaryHead"], "[", + DynamicModuleBox[{ Typeset`open$$ = False, Typeset`embedState$$ = "Ready"}, TemplateBox[{ @@ -10093,13 +10293,11 @@ VbrxdPOhWy/d9aS7X6R9J+HqfwAOKbDD Magnification])}]], GridBox[{{ RowBox[{ - TagBox[ - "\"Number of Transformations: \"", - "SummaryItemAnnotation"], "\[InvisibleSpace]", + TagBox["\"Transforms: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", TagBox["5", "SummaryItem"]}]}, { RowBox[{ - TagBox[ - "\"Dimensions of each: \"", "SummaryItemAnnotation"], + TagBox["\"Dimensions: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox[ RowBox[{"{", @@ -10131,19 +10329,17 @@ VbrxdPOhWy/d9aS7X6R9J+HqfwAOKbDD Magnification])}]], GridBox[{{ RowBox[{ - TagBox[ - "\"Number of Transformations: \"", - "SummaryItemAnnotation"], "\[InvisibleSpace]", + TagBox["\"Transforms: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", TagBox["5", "SummaryItem"]}]}, { RowBox[{ - TagBox[ - 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{2, + 13} -> Complex[0.11666092373220538`, 0.218936695047531], { + 2, 14} -> + Complex[-0.0868392972934217, -0.014513008624828502`], {2, + 15} -> Complex[ + 0.01747597846754094, -0.10203270358947614`], {2, 16} -> + Complex[-0.2917595352077467, -0.3784536838475334], {2, + 17} -> Complex[-0.23097227051538433`, + 0.42320695740719316`], {2, 18} -> + Complex[-0.16418162796222252`, -0.13133580222113328`], {2, + 19} -> + Complex[-0.06797212261961828, -0.14596193524618625`], {2, + 20} -> Complex[ + 0.029019428968993538`, -0.21528502386671927`], {2, 21} -> + Complex[-0.2782895722096578, 0.5061420791848172], {2, 22} -> + Complex[-0.06947320790243908, -0.12537107773431694`], {2, + 23} -> 1, {3, 4} -> + Complex[0.12373001666450631`, 0.], {3, 5} -> + Complex[-0.029720518023860276`, 0.053544088868121745`], { + 3, 6} -> Complex[-0.15211930857597808`, + 0.0462428201376506], {3, 7} -> + Complex[-0.13167457410588743`, 0.003054508743256395], {3, + 8} -> Complex[ + 0.053161902759652525`, 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0.], {3, 22} -> + Complex[0.12373001666450631`, 0.], {3, 23} -> + Complex[-0.06947320790243908, 0.12537107773431694`], {3, + 24} -> Complex[-0.1766768302665536, -0.1352957631030186], \ +{4, 5} -> Complex[0.030636763457055183`, 0.11355874224143539`], {4, 6} -> + Complex[-0.5748910514001031, 0.14633045383134305`], {4, + 7} -> Complex[ + 0.037739555243020126`, 0.012308050579545225`], {4, 8} -> + Complex[0.2616855917631509, 0.009667937264448562], {4, 9} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {4, + 10} -> Complex[-0.2346066684020043, + 0.5335785389713812], {4, 11} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {4, 12} -> Complex[ - 0.14142597329959572`, -0.061350391806736695`], {6, 13} -> - Complex[-0.04214475287108173, 0.2125642839376449], {6, + 0.25833102475550235`, 0.37839808713437206`], {4, 13} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {4, 14} -> Complex[ - 0.14142597329959572`, -0.061350391806736695`], {6, 15} -> - Complex[-0.08181885113853848, -0.0223150746441549], {6, + 0.25833102475550235`, 0.37839808713437206`], {4, 15} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {4, + 16} -> Complex[-0.2346066684020043, + 0.5335785389713812], {4, 17} -> + Complex[0.037739555243020126`, 0.012308050579545225`], {4, + 18} -> Complex[ + 0.2616855917631509, 0.009667937264448562], {4, 19} -> + Complex[0.030636763457055183`, 0.11355874224143539`], {4, + 20} -> Complex[-0.5748910514001031, + 0.14633045383134305`], {4, 21} -> + Complex[0.8762699833354937, 0.], {4, 22} -> + Complex[-1.1102230246251565`*^-16, 0.], {4, 23} -> + Complex[-0.2782895722096578, -0.5061420791848172], {4, + 24} -> Complex[-0.41976082253431385`, + 0.08596477743313696], {5, 6} -> + Complex[0.7835951260215941, 0.], {5, 7} -> + Complex[0.2578656808252187, -0.10876030801336257`], {5, + 8} -> Complex[-0.17834040611966948`, \ +-0.12609948481270494`], {5, 9} -> + Complex[0.13476314496413244`, 0.0767423516894553], {5, + 10} -> Complex[ + 0.12289396168675226`, -0.38209796458606343`], {5, 11} -> + Complex[-0.19299966150123138`, 0.3282026155242766], {5, + 12} -> Complex[-0.3239368353423375, \ +-0.27853835234466173`], {5, 13} -> + Complex[-0.19299966150123138`, 0.3282026155242766], {5, + 14} -> Complex[-0.3239368353423375, \ +-0.27853835234466173`], {5, 15} -> + Complex[0.13476314496413244`, 0.0767423516894553], {5, 16} -> Complex[ - 0.22168267144312054`, 0.03299920735716682], {6, 17} -> - Complex[-0.02707098756043888, 0.08415309369657159], {6, - 18} -> Complex[-0.03764992141799037, \ --0.13605099778794436`], {6, 19} -> Complex[0.460087813322552, 0.], {6, 20} -> - Complex[-0.27467400107696954`, -0.08314412135251599], {6, - 21} -> Complex[-0.020878723063805726`, - 0.1044611112500656], {6, 22} -> - Complex[0.24014852879671952`, -0.15329539227786965`], {6, - 23} -> Complex[ - 0.11519091687891489`, -0.4012928958026726], {6, 24} -> - Complex[-0.45588071484204273`, 0.014752170920114424`], {7, - 8} -> Complex[0.09630860197729411, 0.], {7, 9} -> - Complex[0.06911881436797135, 0.04269376157510484], {7, - 10} -> Complex[-0.11463533571593523`, - 0.055210078698593804`], {7, 11} -> - Complex[-0.06875971685231415, 0.006550328119388341], {7, - 12} -> Complex[-0.010533198723196371`, \ --0.006296053518233197], {7, 13} -> - Complex[-0.06875971685231415, 0.006550328119388341], {7, - 14} -> Complex[-0.010533198723196371`, \ --0.006296053518233197], {7, 15} -> - Complex[0.06911881436797135, 0.04269376157510484], {7, - 16} -> Complex[-0.11463533571593523`, - 0.055210078698593804`], {7, 17} -> - Complex[-0.13812244808851923`, 0.022678210677030422`], {7, - 18} -> Complex[0.09630860197729411, 0.], {7, 19} -> - Complex[-0.03764992141799037, 0.13605099778794436`], {7, + 0.12289396168675226`, -0.38209796458606343`], {5, 17} -> + Complex[0.2578656808252187, -0.10876030801336257`], {5, + 18} -> Complex[-0.17834040611966948`, \ +-0.12609948481270494`], {5, 19} -> + Complex[ + 8.326672684688674*^-17, -1.3877787807814457`*^-17], {5, + 20} -> Complex[0.7835951260215941, 0.], {5, 21} -> + Complex[-0.5748910514001031, -0.14633045383134305`], {5, + 22} -> Complex[-0.15211930857597808`, \ +-0.0462428201376506], {5, 23} -> + Complex[0.029019428968993538`, 0.21528502386671927`], {5, + 24} -> Complex[ + 0.46472596940822686`, 0.06199143840307167], {6, 7} -> + Complex[-0.007671097858546656, -0.0057340892009139326`], { + 6, 8} -> Complex[ + 0.22516518597054483`, -0.1013577129442031], {6, 9} -> + Complex[0.02081012384838236, 0.14234469229568147`], {6, + 10} -> Complex[-0.09460977915945387, + 0.11933166053315665`], {6, 11} -> + Complex[0.017368709836144688`, 0.1497980533645717], {6, + 12} -> Complex[ + 0.13688251203715338`, 0.1036078618111689], {6, 13} -> + Complex[0.017368709836144688`, 0.1497980533645717], {6, + 14} -> Complex[ + 0.13688251203715338`, 0.1036078618111689], {6, 15} -> + Complex[0.02081012384838236, 0.14234469229568147`], {6, + 16} -> Complex[-0.09460977915945387, + 0.11933166053315665`], {6, 17} -> + Complex[-0.007671097858546656, -0.0057340892009139326`], { + 6, 18} -> Complex[ + 0.22516518597054483`, -0.1013577129442031], {6, 19} -> + Complex[0.21640487397840627`, 0.], {6, 20} -> + Complex[8.326672684688674*^-17, + 1.3877787807814457`*^-17], {6, 21} -> + Complex[0.030636763457055183`, -0.11355874224143539`], {6, + 22} -> + Complex[-0.029720518023860276`, -0.053544088868121745`], { + 6, 23} -> Complex[-0.06797212261961828, + 0.14596193524618625`], {6, 24} -> + Complex[-0.1599463611986579, 0.3186964622408383], {7, 8} -> + Complex[0.47738623440613004`, 0.], {7, 9} -> + Complex[0.08485061223426596, 0.11980477813807357`], {7, + 10} -> Complex[-0.17479296155780177`, + 0.14732776197358502`], {7, 11} -> + Complex[-0.04249050665712762, -0.090268967382963], {7, + 12} -> Complex[ + 0.09710709374348331, 0.20541302638200343`], {7, 13} -> + Complex[-0.04249050665712762, -0.090268967382963], {7, + 14} -> Complex[ + 0.09710709374348331, 0.20541302638200343`], {7, 15} -> + Complex[0.08485061223426596, 0.11980477813807357`], {7, + 16} -> Complex[-0.17479296155780177`, + 0.14732776197358502`], {7, 17} -> + Complex[-1.7867651802561116`*^-16, \ +-1.1449174941446927`*^-16], {7, 18} -> + Complex[0.47738623440613004`, 0.], {7, 19} -> + Complex[0.22516518597054483`, 0.1013577129442031], {7, + 20} -> Complex[-0.17834040611966948`, + 0.12609948481270494`], {7, 21} -> + Complex[0.2616855917631509, -0.009667937264448562], {7, + 22} -> Complex[ + 0.053161902759652525`, -0.14451247653181823`], {7, 23} -> + Complex[-0.16418162796222252`, 0.13133580222113328`], {7, + 24} -> Complex[-0.42262163352679005`, + 0.31456841745062747`], {8, 9} -> + Complex[0.17479296155780188`, 0.14732776197358494`], {8, + 10} -> Complex[-0.08485061223426603, + 0.11980477813807351`], {8, 11} -> + Complex[0.11065480592987563`, -0.022498803837048064`], {8, + 12} -> + Complex[-0.13183382282185513`, -0.15967490110950291`], {8, + 13} -> Complex[ + 0.11065480592987563`, -0.022498803837048064`], {8, 14} -> + Complex[-0.13183382282185513`, -0.15967490110950291`], {8, + 15} -> Complex[ + 0.17479296155780188`, 0.14732776197358494`], {8, 16} -> + Complex[-0.08485061223426603, 0.11980477813807351`], {8, + 17} -> Complex[0.5226137655938699, 0.], {8, 18} -> + Complex[-1.7867651802561116`*^-16, + 1.1449174941446927`*^-16], {8, 19} -> + Complex[-0.007671097858546656, 0.0057340892009139326`], { + 8, 20} -> Complex[ + 0.2578656808252187, 0.10876030801336257`], {8, 21} -> + Complex[0.037739555243020126`, -0.012308050579545225`], { + 8, 22} -> + Complex[-0.13167457410588743`, -0.003054508743256395], {8, + 23} -> + Complex[-0.23097227051538433`, -0.42320695740719316`], {8, + 24} -> Complex[ + 0.29687889692943875`, 0.2375326567068352], {9, 10} -> + Complex[0.619974902703056, 0.], {9, 11} -> + Complex[-0.1905043599805688, -0.13762822610920328`], {9, + 12} -> Complex[ + 0.13800495411149877`, -0.3680867713233184], {9, 13} -> + Complex[-0.1905043599805688, -0.13762822610920328`], {9, + 14} -> Complex[ + 0.13800495411149877`, -0.3680867713233184], {9, 15} -> + Complex[-8.673617379884035*^-17, + 8.326672684688674*^-17], {9, 16} -> + Complex[0.619974902703056, 0.], {9, 17} -> + Complex[-0.08485061223426603, -0.11980477813807351`], {9, + 18} -> Complex[-0.17479296155780177`, \ +-0.14732776197358502`], {9, 19} -> + Complex[-0.09460977915945387, -0.11933166053315665`], {9, 20} -> Complex[ - 0.12702736104247997`, -0.01646149694625006], {7, 21} -> - Complex[-0.07115780937991523, -0.011205895451017422`], {7, - 22} -> Complex[ - 0.07416506719537758, 0.08051189278847616], {7, 23} -> - Complex[0.16774613988004144`, 0.225655277072902], {7, 24} -> - Complex[-0.09714752480970831, -0.05391368675976281], {8, - 9} -> Complex[ - 0.14880448551485936`, -0.18343023977218464`], {8, 10} -> - Complex[0.1868664022204954, 0.06385028540854319], {8, 11} -> - Complex[0.4351121808870284, -0.1282880779202612], {8, 12} -> - Complex[-0.2300764357164763, 0.11137567802408056`], {8, - 13} -> Complex[0.4351121808870284, -0.1282880779202612], { - 8, 14} -> Complex[-0.2300764357164763, - 0.11137567802408056`], {8, 15} -> - Complex[0.14880448551485936`, -0.18343023977218464`], {8, - 16} -> Complex[0.1868664022204954, 0.06385028540854319], { - 8, 17} -> Complex[0.7694387170582185, 0.], {8, 18} -> - Complex[-0.13812244808851923`, -0.022678210677030422`], { - 8, 19} -> - Complex[-0.02707098756043888, -0.08415309369657159], {8, - 20} -> Complex[-0.4024303768625533, \ --0.06319936655428628], {8, 21} -> - Complex[0.6164020052722983, 0.03301684548082712], {8, 22} -> - Complex[-0.2385656438458264, -0.06903700176549488], {8, - 23} -> Complex[-0.24737746704016716`, \ --0.5022802985841684], {8, 24} -> - Complex[0.199408104684111, 0.11720650121625686`], {9, 10} -> - Complex[0.41416215192864014`, 0.], {9, 11} -> - Complex[-0.1634450723422762, -0.0011076658174966336`], {9, - 12} -> Complex[ - 0.24390216997712094`, 0.12967162455970677`], {9, 13} -> - Complex[-0.1634450723422762, -0.0011076658174966336`], {9, - 14} -> Complex[ - 0.24390216997712094`, 0.12967162455970677`], {9, 15} -> - Complex[-0.29089675324323006`, -0.1558046632134409], {9, - 16} -> Complex[0.41416215192864014`, 0.], {9, 17} -> - Complex[0.1868664022204954, -0.06385028540854319], {9, - 18} -> Complex[-0.11463533571593523`, \ --0.055210078698593804`], {9, 19} -> - Complex[0.22168267144312054`, -0.03299920735716682], {9, - 20} -> Complex[-0.2934854494126851, \ --0.04085502210736608], {9, 21} -> - Complex[-0.06237372995183321, 0.06727056435677253], {9, - 22} -> Complex[-0.030346695869449408`, \ --0.030149558835644692`], {9, 23} -> - Complex[-0.09727281618814543, -0.4251726947214757], {9, - 24} -> Complex[0.02023138254715448, -0.11137525641609], { + 0.12289396168675226`, 0.38209796458606343`], {9, 21} -> + Complex[-0.2346066684020043, -0.5335785389713812], {9, + 22} -> Complex[ + 0.08210404404737587, 0.05307307167276523], {9, 23} -> + Complex[-0.2917595352077467, 0.3784536838475334], {9, 24} -> + Complex[0.33333532926317844`, 0.0052698415932301845`], { 10, 11} -> - Complex[0.3689066714841379, -0.01963818646965646], {10, - 12} -> - Complex[-0.364332505630415, -0.055629112414147225`], {10, - 13} -> Complex[ - 0.3689066714841379, -0.01963818646965646], {10, 14} -> - Complex[-0.364332505630415, -0.055629112414147225`], {10, - 15} -> Complex[0.4159087185308232, 0.], {10, 16} -> - Complex[-0.29089675324323006`, 0.1558046632134409], {10, + Complex[0.1899783589314758, -0.1365082746823964], {10, + 12} -> Complex[-0.15558917647409173`, \ +-0.09979098803090053], {10, 13} -> + Complex[0.1899783589314758, -0.1365082746823964], {10, + 14} -> Complex[-0.15558917647409173`, \ +-0.09979098803090053], {10, 15} -> Complex[0.3800250972969439, 0.], {10, 16} -> + Complex[-8.673617379884035*^-17, \ +-8.326672684688674*^-17], {10, 17} -> + Complex[0.17479296155780188`, -0.14732776197358494`], {10, + 18} -> Complex[ + 0.08485061223426596, -0.11980477813807357`], {10, 19} -> + Complex[0.02081012384838236, -0.14234469229568147`], {10, + 20} -> Complex[ + 0.13476314496413244`, -0.0767423516894553], {10, 21} -> + Complex[-0.15157725194981503`, 0.17844414940708186`], {10, + 22} -> + Complex[-0.04368290380765367, -0.027436459786564067`], { + 10, 23} -> + Complex[0.01747597846754094, 0.10203270358947614`], {10, + 24} -> Complex[ + 0.34087862056969404`, 0.1100388283919505], {11, 12} -> + Complex[0.4724699088969763, 0.], {11, 13} -> + Complex[ + 1.8041124150158796`*^-16, -1.457167719820518*^-16], {11, + 14} -> Complex[0.4724699088969763, 0.], {11, 15} -> + Complex[-0.15558917647409173`, 0.09979098803090053], {11, + 16} -> Complex[ + 0.13800495411149877`, 0.3680867713233184], {11, 17} -> + Complex[-0.13183382282185513`, 0.15967490110950291`], {11, + 18} -> Complex[ + 0.09710709374348331, -0.20541302638200343`], {11, 19} -> + Complex[0.13688251203715338`, -0.1036078618111689], {11, + 20} -> Complex[-0.3239368353423375, + 0.27853835234466173`], {11, 21} -> + Complex[0.25833102475550235`, -0.37839808713437206`], {11, + 22} -> Complex[ + 0.01476498538421066, -0.0004105289599095289], {11, 23} -> + Complex[-0.0868392972934217, 0.014513008624828502`], {11, + 24} -> Complex[-0.16585075463443222`, + 0.4085141858532183], {12, 13} -> + Complex[0.5275300911030236, 0.], {12, 14} -> + Complex[1.8041124150158796`*^-16, + 1.457167719820518*^-16], {12, 15} -> + Complex[0.1899783589314758, 0.1365082746823964], {12, 16} -> + Complex[-0.1905043599805688, 0.13762822610920328`], {12, 17} -> Complex[ - 0.14880448551485936`, 0.18343023977218464`], {10, 18} -> - Complex[0.06911881436797135, -0.04269376157510484], {10, - 19} -> Complex[-0.08181885113853848, - 0.0223150746441549], {10, 20} -> - Complex[0.0806509155236052, -0.06238441832735686], {10, - 21} -> Complex[ - 0.3146513247958228, -0.026797378998685423`], {10, 22} -> - Complex[0.011359633728347016`, -0.08520800008182422], {10, + 0.11065480592987563`, 0.022498803837048064`], {12, 18} -> + Complex[-0.04249050665712762, 0.090268967382963], {12, + 19} -> Complex[ + 0.017368709836144688`, -0.1497980533645717], {12, 20} -> + Complex[-0.19299966150123138`, -0.3282026155242766], {12, + 21} -> Complex[-0.06250476454855013, + 0.34969666557353624`], {12, 22} -> + Complex[-0.008146862343863803, 0.10273781027166022`], {12, 23} -> Complex[ - 0.17837001887568327`, 0.09187489939370964], {10, 24} -> - Complex[-0.07164664525064873, 0.2907134789193454], {11, - 12} -> Complex[0.3744935743083003, 0.], {11, 13} -> - Complex[-0.4009452410407828, 0.11284019268466222`], {11, - 14} -> Complex[0.3744935743083003, 0.], {11, 15} -> - Complex[-0.364332505630415, 0.055629112414147225`], {11, + 0.11666092373220538`, -0.218936695047531], {12, 24} -> + Complex[0.161016937967343, 0.024863370726720582`], {13, + 14} -> Complex[0.4724699088969763, 0.], {13, 15} -> + Complex[-0.15558917647409173`, 0.09979098803090053], {13, 16} -> Complex[ - 0.24390216997712094`, -0.12967162455970677`], {11, 17} -> - 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0.011205895451017422`], {4, 19} -> - Complex[-0.020878723063805726`, -0.1044611112500656], {4, 20} -> - Complex[-0.32198385415100245`, 0.056214448752933965`], {4, 21} -> - Complex[0.7322837744378387, 0.], {4, 22} -> - Complex[-0.1157253008903192, -0.07182611485724257], {4, 23} -> - Complex[-0.29340373279675197`, -0.27809938962030384`], {4, 24} -> - Complex[-0.07968743240634073, 0.17702649633258472`], {5, 6} -> - Complex[0.5399121866774477, 0.], {5, 7} -> - Complex[-0.4024303768625533, 0.06319936655428628], {5, 8} -> - Complex[0.12702736104247997`, 0.01646149694625006], {5, 9} -> - Complex[0.0806509155236052, 0.06238441832735686], {5, 10} -> - Complex[-0.2934854494126851, 0.04085502210736608], {5, 11} -> - Complex[-0.18425433538775948`, -0.03665899040288985], {5, 12} -> - Complex[-0.029186234486845222`, -0.022372160811117286`], {5, 13} -> - Complex[-0.18425433538775948`, -0.03665899040288985], {5, 14} -> - Complex[-0.029186234486845222`, -0.022372160811117286`], {5, 15} -> - Complex[0.0806509155236052, 0.06238441832735686], {5, 16} -> - Complex[-0.2934854494126851, 0.04085502210736608], {5, 17} -> - Complex[-0.4024303768625533, 0.06319936655428628], {5, 18} -> - Complex[0.12702736104247997`, 0.01646149694625006], {5, 19} -> - Complex[-0.27467400107696954`, 0.08314412135251599], {5, 20} -> - Complex[0.5399121866774477, 0.], {5, 21} -> - Complex[-0.32198385415100245`, -0.056214448752933965`], {5, 22} -> - Complex[0.02087872306380592, 0.10446111125006569`], {5, 23} -> - Complex[0.34202714802636813`, 0.5203828231210746], {5, 24} -> - Complex[0.27130379014783124`, 0.12044715596755934`], {6, 7} -> - Complex[-0.02707098756043888, 0.08415309369657159], {6, 8} -> - Complex[-0.03764992141799037, -0.13605099778794436`], {6, 9} -> - Complex[-0.08181885113853848, -0.0223150746441549], {6, 10} -> - Complex[0.22168267144312054`, 0.03299920735716682], {6, 11} -> - Complex[-0.04214475287108173, 0.2125642839376449], {6, 12} -> - Complex[0.14142597329959572`, -0.061350391806736695`], {6, 13} -> - Complex[-0.04214475287108173, 0.2125642839376449], {6, 14} -> - Complex[0.14142597329959572`, -0.061350391806736695`], {6, 15} -> - Complex[-0.08181885113853848, -0.0223150746441549], {6, 16} -> - Complex[0.22168267144312054`, 0.03299920735716682], {6, 17} -> - Complex[-0.02707098756043888, 0.08415309369657159], {6, 18} -> - Complex[-0.03764992141799037, -0.13605099778794436`], {6, 19} -> - Complex[0.460087813322552, 0.], {6, 20} -> - Complex[-0.27467400107696954`, -0.08314412135251599], {6, 21} -> - Complex[-0.020878723063805726`, 0.1044611112500656], {6, 22} -> - Complex[0.24014852879671952`, -0.15329539227786965`], {6, 23} -> - Complex[0.11519091687891489`, -0.4012928958026726], {6, 24} -> - Complex[-0.45588071484204273`, 0.014752170920114424`], {7, 8} -> - Complex[0.09630860197729411, 0.], {7, 9} -> - Complex[0.06911881436797135, 0.04269376157510484], {7, 10} -> - Complex[-0.11463533571593523`, 0.055210078698593804`], {7, 11} -> - Complex[-0.06875971685231415, 0.006550328119388341], {7, 12} -> - Complex[-0.010533198723196371`, -0.006296053518233197], {7, 13} -> - Complex[-0.06875971685231415, 0.006550328119388341], {7, 14} -> - Complex[-0.010533198723196371`, -0.006296053518233197], {7, 15} -> - Complex[0.06911881436797135, 0.04269376157510484], {7, 16} -> - Complex[-0.11463533571593523`, 0.055210078698593804`], {7, 17} -> - Complex[-0.13812244808851923`, 0.022678210677030422`], {7, 18} -> - Complex[0.09630860197729411, 0.], {7, 19} -> - Complex[-0.03764992141799037, 0.13605099778794436`], {7, 20} -> - Complex[0.12702736104247997`, -0.01646149694625006], {7, 21} -> - Complex[-0.07115780937991523, -0.011205895451017422`], {7, 22} -> - Complex[0.07416506719537758, 0.08051189278847616], {7, 23} -> - Complex[0.16774613988004144`, 0.225655277072902], {7, 24} -> - Complex[-0.09714752480970831, -0.05391368675976281], {8, 9} -> - Complex[0.14880448551485936`, -0.18343023977218464`], {8, 10} -> - Complex[0.1868664022204954, 0.06385028540854319], {8, 11} -> - Complex[0.4351121808870284, -0.1282880779202612], {8, 12} -> - Complex[-0.2300764357164763, 0.11137567802408056`], {8, 13} -> - Complex[0.4351121808870284, -0.1282880779202612], {8, 14} -> - Complex[-0.2300764357164763, 0.11137567802408056`], {8, 15} -> - Complex[0.14880448551485936`, -0.18343023977218464`], {8, 16} -> - Complex[0.1868664022204954, 0.06385028540854319], {8, 17} -> - Complex[0.7694387170582185, 0.], {8, 18} -> - Complex[-0.13812244808851923`, -0.022678210677030422`], {8, 19} -> - Complex[-0.02707098756043888, -0.08415309369657159], {8, 20} -> - Complex[-0.4024303768625533, -0.06319936655428628], {8, 21} -> - Complex[0.6164020052722983, 0.03301684548082712], {8, 22} -> - Complex[-0.2385656438458264, -0.06903700176549488], {8, 23} -> - Complex[-0.24737746704016716`, -0.5022802985841684], {8, 24} -> - Complex[0.199408104684111, 0.11720650121625686`], {9, 10} -> - Complex[0.41416215192864014`, 0.], {9, 11} -> - Complex[-0.1634450723422762, -0.0011076658174966336`], {9, 12} -> - Complex[0.24390216997712094`, 0.12967162455970677`], {9, 13} -> - Complex[-0.1634450723422762, -0.0011076658174966336`], {9, 14} -> - Complex[0.24390216997712094`, 0.12967162455970677`], {9, 15} -> - Complex[-0.29089675324323006`, -0.1558046632134409], {9, 16} -> - Complex[0.41416215192864014`, 0.], {9, 17} -> - Complex[0.1868664022204954, -0.06385028540854319], {9, 18} -> - Complex[-0.11463533571593523`, -0.055210078698593804`], {9, 19} -> - Complex[0.22168267144312054`, -0.03299920735716682], {9, 20} -> - Complex[-0.2934854494126851, -0.04085502210736608], {9, 21} -> - Complex[-0.06237372995183321, 0.06727056435677253], {9, 22} -> - Complex[-0.030346695869449408`, -0.030149558835644692`], {9, 23} -> - Complex[-0.09727281618814543, -0.4251726947214757], {9, 24} -> - Complex[0.02023138254715448, -0.11137525641609], {10, 11} -> - Complex[0.3689066714841379, -0.01963818646965646], {10, 12} -> - Complex[-0.364332505630415, -0.055629112414147225`], {10, 13} -> - Complex[0.3689066714841379, -0.01963818646965646], {10, 14} -> - Complex[-0.364332505630415, -0.055629112414147225`], {10, 15} -> - Complex[0.4159087185308232, 0.], {10, 16} -> - Complex[-0.29089675324323006`, 0.1558046632134409], {10, 17} -> - Complex[0.14880448551485936`, 0.18343023977218464`], {10, 18} -> - Complex[0.06911881436797135, -0.04269376157510484], {10, 19} -> - Complex[-0.08181885113853848, 0.0223150746441549], {10, 20} -> - Complex[0.0806509155236052, -0.06238441832735686], {10, 21} -> - Complex[0.3146513247958228, -0.026797378998685423`], {10, 22} -> - Complex[0.011359633728347016`, -0.08520800008182422], {10, 23} -> - Complex[0.17837001887568327`, 0.09187489939370964], {10, 24} -> - Complex[-0.07164664525064873, 0.2907134789193454], {11, 12} -> - Complex[0.3744935743083003, 0.], {11, 13} -> - Complex[-0.4009452410407828, 0.11284019268466222`], {11, 14} -> - Complex[0.3744935743083003, 0.], {11, 15} -> - Complex[-0.364332505630415, 0.055629112414147225`], {11, 16} -> - Complex[0.24390216997712094`, -0.12967162455970677`], {11, 17} -> - Complex[-0.2300764357164763, -0.11137567802408056`], {11, 18} -> - Complex[-0.010533198723196371`, 0.006296053518233197], {11, 19} -> - Complex[0.14142597329959572`, 0.061350391806736695`], {11, 20} -> - Complex[-0.029186234486845222`, 0.022372160811117286`], {11, 21} -> - Complex[-0.33684082460892883`, 0.09005315780013691], {11, 22} -> - Complex[0.09059534447844704, 0.10345647613127328`], {11, 23} -> - Complex[-0.008851768247553363, -0.029068202106085572`], {11, 24} -> - Complex[-0.07795077274116374, -0.2904671042221048], {12, 13} -> - Complex[0.5794387931418044, 0.], {12, 14} -> - Complex[-0.4009452410407828, -0.11284019268466222`], {12, 15} -> - Complex[0.3689066714841379, 0.01963818646965646], {12, 16} -> - Complex[-0.1634450723422762, 0.0011076658174966336`], {12, 17} -> - Complex[0.4351121808870284, 0.1282880779202612], {12, 18} -> - Complex[-0.06875971685231415, -0.006550328119388341], {12, 19} -> - Complex[-0.04214475287108173, -0.2125642839376449], {12, 20} -> - Complex[-0.18425433538775948`, 0.03665899040288985], {12, 21} -> - Complex[0.5691291778553188, 0.009546455336403484], {12, 22} -> - Complex[-0.11604982216622163`, -0.19926182212673743`], {12, 23} -> - Complex[-0.20476687273928368`, -0.20940927658482447`], {12, 24} -> - Complex[-0.04633762187626639, 0.37156097014999256`], {13, 14} -> - Complex[0.3744935743083003, 0.], {13, 15} -> - Complex[-0.364332505630415, 0.055629112414147225`], {13, 16} -> - Complex[0.24390216997712094`, -0.12967162455970677`], {13, 17} -> - Complex[-0.2300764357164763, -0.11137567802408056`], {13, 18} -> - Complex[-0.010533198723196371`, 0.006296053518233197], {13, 19} -> - Complex[0.14142597329959572`, 0.061350391806736695`], {13, 20} -> - Complex[-0.029186234486845222`, 0.022372160811117286`], {13, 21} -> - Complex[-0.33684082460892883`, 0.09005315780013691], {13, 22} -> - Complex[0.09059534447844704, 0.10345647613127328`], {13, 23} -> - Complex[-0.008851768247553363, -0.029068202106085572`], {13, 24} -> - Complex[-0.07795077274116374, -0.2904671042221048], {14, 15} -> - Complex[0.3689066714841379, 0.01963818646965646], {14, 16} -> - Complex[-0.1634450723422762, 0.0011076658174966336`], {14, 17} -> - Complex[0.4351121808870284, 0.1282880779202612], {14, 18} -> - Complex[-0.06875971685231415, -0.006550328119388341], {14, 19} -> - Complex[-0.04214475287108173, -0.2125642839376449], {14, 20} -> - Complex[-0.18425433538775948`, 0.03665899040288985], {14, 21} -> - Complex[0.5691291778553188, 0.009546455336403484], {14, 22} -> - Complex[-0.11604982216622163`, -0.19926182212673743`], {14, 23} -> - Complex[-0.20476687273928368`, -0.20940927658482447`], {14, 24} -> - Complex[-0.04633762187626639, 0.37156097014999256`], {15, 16} -> - Complex[0.41416215192864014`, 0.], {15, 17} -> - Complex[0.1868664022204954, -0.06385028540854319], {15, 18} -> - Complex[-0.11463533571593523`, -0.055210078698593804`], {15, 19} -> - Complex[0.22168267144312054`, -0.03299920735716682], {15, 20} -> - Complex[-0.2934854494126851, -0.04085502210736608], {15, 21} -> - Complex[-0.06237372995183321, 0.06727056435677253], {15, 22} -> - Complex[-0.030346695869449408`, -0.030149558835644692`], {15, 23} -> - Complex[-0.09727281618814543, -0.4251726947214757], {15, 24} -> - Complex[0.02023138254715448, -0.11137525641609], {16, 17} -> - Complex[0.14880448551485936`, 0.18343023977218464`], {16, 18} -> - Complex[0.06911881436797135, -0.04269376157510484], {16, 19} -> - Complex[-0.08181885113853848, 0.0223150746441549], {16, 20} -> - Complex[0.0806509155236052, -0.06238441832735686], {16, 21} -> - Complex[0.3146513247958228, -0.026797378998685423`], {16, 22} -> - Complex[0.011359633728347016`, -0.08520800008182422], {16, 23} -> - Complex[0.17837001887568327`, 0.09187489939370964], {16, 24} -> - Complex[-0.07164664525064873, 0.2907134789193454], {17, 18} -> - Complex[0.09630860197729411, 0.], {17, 19} -> - Complex[-0.03764992141799037, 0.13605099778794436`], {17, 20} -> - Complex[0.12702736104247997`, -0.01646149694625006], {17, 21} -> - Complex[-0.07115780937991523, -0.011205895451017422`], {17, 22} -> - Complex[0.07416506719537758, 0.08051189278847616], {17, 23} -> - Complex[0.16774613988004144`, 0.225655277072902], {17, 24} -> - Complex[-0.09714752480970831, -0.05391368675976281], {18, 19} -> - Complex[-0.02707098756043888, -0.08415309369657159], {18, 20} -> - Complex[-0.4024303768625533, -0.06319936655428628], {18, 21} -> - Complex[0.6164020052722983, 0.03301684548082712], {18, 22} -> - Complex[-0.2385656438458264, -0.06903700176549488], {18, 23} -> - Complex[-0.24737746704016716`, -0.5022802985841684], {18, 24} -> - Complex[0.199408104684111, 0.11720650121625686`], {19, 20} -> - Complex[0.5399121866774477, 0.], {19, 21} -> - Complex[-0.32198385415100245`, -0.056214448752933965`], {19, 22} -> - Complex[0.02087872306380592, 0.10446111125006569`], {19, 23} -> - Complex[0.34202714802636813`, 0.5203828231210746], {19, 24} -> - Complex[0.27130379014783124`, 0.12044715596755934`], {20, 21} -> - Complex[-0.020878723063805726`, 0.1044611112500656], {20, 22} -> - Complex[0.24014852879671952`, -0.15329539227786965`], {20, 23} -> - Complex[0.11519091687891489`, -0.4012928958026726], {20, 24} -> - Complex[-0.45588071484204273`, 0.014752170920114424`], {21, 22} -> - Complex[0.2677162255621613, 0.], {21, 23} -> - Complex[0.27244326195988916`, -0.011111878577649018`], {21, 24} -> - Complex[-0.3506829757791454, -0.10726001244941838`], {22, 23} -> - Complex[-0.29340373279675197`, -0.27809938962030384`], {22, 24} -> - Complex[-0.07968743240634073, 0.17702649633258472`]}, + Complex[-0.2782895722096578, 0.5061420791848172], {2, 4} -> + Complex[-0.06947320790243908, -0.12537107773431694`], {2, 5} -> + Complex[-0.06797212261961828, -0.14596193524618625`], {2, 6} -> + Complex[0.029019428968993538`, -0.21528502386671927`], {2, 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+ 0.12609948481270494`], {8, 21} -> + Complex[0.2616855917631509, -0.009667937264448562], {8, 22} -> Complex[ - 0.24014852879671952`, -0.15329539227786965`], {5, 23} -> - Complex[0.11519091687891489`, -0.4012928958026726], {5, - 24} -> Complex[-0.45588071484204273`, - 0.014752170920114424`], {6, 7} -> - Complex[-0.4024303768625533, 0.06319936655428628], {6, 8} -> - Complex[0.12702736104247997`, 0.01646149694625006], {6, - 9} -> Complex[-0.2934854494126851, 0.04085502210736608], { - 6, 10} -> Complex[ - 0.0806509155236052, 0.06238441832735686], {6, 11} -> - Complex[-0.029186234486845222`, -0.022372160811117286`], { - 6, 12} -> - Complex[-0.18425433538775948`, -0.03665899040288985], {6, - 13} -> Complex[-0.029186234486845222`, \ --0.022372160811117286`], {6, 14} -> - Complex[-0.18425433538775948`, -0.03665899040288985], {6, - 15} -> Complex[-0.2934854494126851, - 0.04085502210736608], {6, 16} -> - Complex[0.0806509155236052, 0.06238441832735686], {6, 17} -> - Complex[-0.4024303768625533, 0.06319936655428628], {6, + 0.053161902759652525`, -0.14451247653181823`], {8, 23} -> + Complex[-0.16418162796222252`, 0.13133580222113328`], {8, + 24} -> Complex[-0.42262163352679005`, + 0.31456841745062747`], {9, 10} -> + Complex[0.3800250972969439, 0.], {9, 11} -> + Complex[0.1899783589314758, -0.1365082746823964], {9, 12} -> + Complex[-0.15558917647409173`, -0.09979098803090053], {9, + 13} -> Complex[0.1899783589314758, -0.1365082746823964], { + 9, 14} -> + Complex[-0.15558917647409173`, -0.09979098803090053], {9, + 15} -> Complex[-8.673617379884035*^-17, \ +-8.326672684688674*^-17], {9, 16} -> Complex[0.3800250972969439, 0.], {9, 17} -> + Complex[0.08485061223426596, -0.11980477813807357`], {9, 18} -> Complex[ - 0.12702736104247997`, 0.01646149694625006], {6, 19} -> - Complex[0.5399121866774477, 0.], {6, 20} -> - Complex[-0.27467400107696954`, 0.08314412135251599], {6, - 21} -> Complex[-0.32198385415100245`, \ --0.056214448752933965`], {6, 22} -> - Complex[0.02087872306380592, 0.10446111125006569`], {6, - 23} -> Complex[ - 0.34202714802636813`, 0.5203828231210746], {6, 24} -> - Complex[0.27130379014783124`, 0.12044715596755934`], {7, - 8} -> Complex[0.09630860197729411, 0.], {7, 9} -> - Complex[-0.11463533571593523`, 0.055210078698593804`], {7, - 10} -> Complex[ - 0.06911881436797135, 0.04269376157510484], {7, 11} -> - Complex[-0.010533198723196371`, -0.006296053518233197], { - 7, 12} -> Complex[-0.06875971685231415, - 0.006550328119388341], {7, 13} -> - Complex[-0.010533198723196371`, -0.006296053518233197], { - 7, 14} -> Complex[-0.06875971685231415, - 0.006550328119388341], {7, 15} -> - Complex[-0.11463533571593523`, 0.055210078698593804`], {7, - 16} -> Complex[ - 0.06911881436797135, 0.04269376157510484], {7, 17} -> - Complex[-0.13812244808851923`, 0.022678210677030422`], {7, - 18} -> Complex[0.09630860197729411, 0.], {7, 19} -> - Complex[0.12702736104247997`, -0.01646149694625006], {7, - 20} -> Complex[-0.03764992141799037, - 0.13605099778794436`], {7, 21} -> - Complex[-0.07115780937991523, -0.011205895451017422`], {7, - 22} -> Complex[ - 0.07416506719537758, 0.08051189278847616], {7, 23} -> - Complex[0.16774613988004144`, 0.225655277072902], {7, 24} -> - Complex[-0.09714752480970831, -0.05391368675976281], {8, - 9} -> Complex[0.1868664022204954, 0.06385028540854319], { - 8, 10} -> Complex[ - 0.14880448551485936`, -0.18343023977218464`], {8, 11} -> - Complex[-0.2300764357164763, 0.11137567802408056`], {8, - 12} -> Complex[0.4351121808870284, -0.1282880779202612], { - 8, 13} -> Complex[-0.2300764357164763, - 0.11137567802408056`], {8, 14} -> - Complex[0.4351121808870284, -0.1282880779202612], {8, 15} -> - Complex[0.1868664022204954, 0.06385028540854319], {8, 16} -> - Complex[0.14880448551485936`, -0.18343023977218464`], {8, - 17} -> Complex[0.7694387170582185, 0.], {8, 18} -> - Complex[-0.13812244808851923`, -0.022678210677030422`], { - 8, 19} -> - Complex[-0.4024303768625533, -0.06319936655428628], {8, - 20} -> Complex[-0.02707098756043888, \ --0.08415309369657159], {8, 21} -> - Complex[0.6164020052722983, 0.03301684548082712], {8, 22} -> - Complex[-0.2385656438458264, -0.06903700176549488], {8, - 23} -> Complex[-0.24737746704016716`, \ --0.5022802985841684], {8, 24} -> - Complex[0.199408104684111, 0.11720650121625686`], {9, 10} -> - Complex[0.4159087185308232, 0.], {9, 11} -> - Complex[-0.364332505630415, -0.055629112414147225`], {9, + 0.17479296155780188`, -0.14732776197358494`], {9, 19} -> + Complex[0.02081012384838236, -0.14234469229568147`], {9, + 20} -> Complex[ + 0.13476314496413244`, -0.0767423516894553], {9, 21} -> + Complex[-0.15157725194981503`, 0.17844414940708186`], {9, + 22} -> Complex[-0.04368290380765367, \ +-0.027436459786564067`], {9, 23} -> + Complex[0.01747597846754094, 0.10203270358947614`], {9, + 24} -> Complex[ + 0.34087862056969404`, 0.1100388283919505], {10, 11} -> + Complex[-0.1905043599805688, -0.13762822610920328`], {10, 12} -> Complex[ - 0.3689066714841379, -0.01963818646965646], {9, 13} -> - Complex[-0.364332505630415, -0.055629112414147225`], {9, + 0.13800495411149877`, -0.3680867713233184], {10, 13} -> + Complex[-0.1905043599805688, -0.13762822610920328`], {10, 14} -> Complex[ - 0.3689066714841379, -0.01963818646965646], {9, 15} -> - Complex[-0.29089675324323006`, 0.1558046632134409], {9, - 16} -> Complex[0.4159087185308232, 0.], {9, 17} -> - Complex[0.14880448551485936`, 0.18343023977218464`], {9, - 18} -> Complex[ - 0.06911881436797135, -0.04269376157510484], {9, 19} -> - Complex[0.0806509155236052, -0.06238441832735686], {9, - 20} -> Complex[-0.08181885113853848, - 0.0223150746441549], {9, 21} -> - Complex[0.3146513247958228, -0.026797378998685423`], {9, + 0.13800495411149877`, -0.3680867713233184], {10, 15} -> + Complex[0.619974902703056, 0.], {10, 16} -> + Complex[-8.673617379884035*^-17, + 8.326672684688674*^-17], {10, 17} -> + Complex[-0.17479296155780177`, -0.14732776197358502`], { + 10, 18} -> + Complex[-0.08485061223426603, -0.11980477813807351`], {10, + 19} -> + Complex[-0.09460977915945387, -0.11933166053315665`], {10, + 20} -> Complex[ + 0.12289396168675226`, 0.38209796458606343`], {10, 21} -> + Complex[-0.2346066684020043, -0.5335785389713812], {10, 22} -> Complex[ - 0.011359633728347016`, -0.08520800008182422], {9, 23} -> - Complex[0.17837001887568327`, 0.09187489939370964], {9, - 24} -> Complex[-0.07164664525064873, - 0.2907134789193454], {10, 11} -> - Complex[0.24390216997712094`, 0.12967162455970677`], {10, - 12} -> Complex[-0.1634450723422762, \ --0.0011076658174966336`], {10, 13} -> - Complex[0.24390216997712094`, 0.12967162455970677`], {10, - 14} -> Complex[-0.1634450723422762, \ --0.0011076658174966336`], {10, 15} -> - Complex[0.41416215192864014`, 0.], {10, 16} -> - Complex[-0.29089675324323006`, -0.1558046632134409], {10, - 17} -> Complex[ - 0.1868664022204954, -0.06385028540854319], {10, 18} -> - Complex[-0.11463533571593523`, -0.055210078698593804`], { - 10, 19} -> - Complex[-0.2934854494126851, -0.04085502210736608], {10, - 20} -> Complex[ - 0.22168267144312054`, -0.03299920735716682], {10, 21} -> - Complex[-0.06237372995183321, 0.06727056435677253], {10, - 22} -> Complex[-0.030346695869449408`, \ --0.030149558835644692`], {10, 23} -> - Complex[-0.09727281618814543, -0.4251726947214757], {10, - 24} -> Complex[0.02023138254715448, -0.11137525641609], { - 11, 12} -> Complex[0.5794387931418044, 0.], {11, 13} -> - Complex[-0.4009452410407828, -0.11284019268466222`], {11, - 14} -> Complex[0.5794387931418044, 0.], {11, 15} -> - Complex[-0.1634450723422762, 0.0011076658174966336`], {11, - 16} -> Complex[ - 0.3689066714841379, 0.01963818646965646], {11, 17} -> - Complex[0.4351121808870284, 0.1282880779202612], {11, 18} -> - Complex[-0.06875971685231415, -0.006550328119388341], {11, - 19} -> Complex[-0.18425433538775948`, - 0.03665899040288985], {11, 20} -> - Complex[-0.04214475287108173, -0.2125642839376449], {11, - 21} -> Complex[ - 0.5691291778553188, 0.009546455336403484], {11, 22} -> - Complex[-0.11604982216622163`, -0.19926182212673743`], { - 11, 23} -> - Complex[-0.20476687273928368`, -0.20940927658482447`], { - 11, 24} -> - Complex[-0.04633762187626639, 0.37156097014999256`], {12, - 13} -> Complex[0.3744935743083003, 0.], {12, 14} -> - Complex[-0.4009452410407828, 0.11284019268466222`], {12, - 15} -> Complex[ - 0.24390216997712094`, -0.12967162455970677`], {12, 16} -> - Complex[-0.364332505630415, 0.055629112414147225`], {12, - 17} -> Complex[-0.2300764357164763, \ --0.11137567802408056`], {12, 18} -> - Complex[-0.010533198723196371`, 0.006296053518233197], { - 12, 19} -> - Complex[-0.029186234486845222`, 0.022372160811117286`], { - 12, 20} -> - Complex[0.14142597329959572`, 0.061350391806736695`], {12, - 21} -> Complex[-0.33684082460892883`, - 0.09005315780013691], {12, 22} -> - Complex[0.09059534447844704, 0.10345647613127328`], {12, - 23} -> Complex[-0.008851768247553363, \ --0.029068202106085572`], {12, 24} -> - Complex[-0.07795077274116374, -0.2904671042221048], {13, - 14} -> Complex[0.5794387931418044, 0.], {13, 15} -> - Complex[-0.1634450723422762, 0.0011076658174966336`], {13, - 16} -> Complex[ - 0.3689066714841379, 0.01963818646965646], {13, 17} -> - Complex[0.4351121808870284, 0.1282880779202612], {13, 18} -> - Complex[-0.06875971685231415, -0.006550328119388341], {13, - 19} -> Complex[-0.18425433538775948`, - 0.03665899040288985], {13, 20} -> - Complex[-0.04214475287108173, -0.2125642839376449], {13, - 21} -> Complex[ - 0.5691291778553188, 0.009546455336403484], {13, 22} -> - Complex[-0.11604982216622163`, -0.19926182212673743`], { - 13, 23} -> - Complex[-0.20476687273928368`, -0.20940927658482447`], { - 13, 24} -> - Complex[-0.04633762187626639, 0.37156097014999256`], {14, - 15} -> Complex[ - 0.24390216997712094`, -0.12967162455970677`], {14, 16} -> - Complex[-0.364332505630415, 0.055629112414147225`], {14, - 17} -> Complex[-0.2300764357164763, \ --0.11137567802408056`], {14, 18} -> - Complex[-0.010533198723196371`, 0.006296053518233197], { - 14, 19} -> - Complex[-0.029186234486845222`, 0.022372160811117286`], { - 14, 20} -> - Complex[0.14142597329959572`, 0.061350391806736695`], {14, - 21} -> Complex[-0.33684082460892883`, - 0.09005315780013691], {14, 22} -> - Complex[0.09059534447844704, 0.10345647613127328`], {14, - 23} -> Complex[-0.008851768247553363, \ --0.029068202106085572`], {14, 24} -> - Complex[-0.07795077274116374, -0.2904671042221048], {15, - 16} -> Complex[0.4159087185308232, 0.], {15, 17} -> - Complex[0.14880448551485936`, 0.18343023977218464`], {15, + 0.08210404404737587, 0.05307307167276523], {10, 23} -> + Complex[-0.2917595352077467, 0.3784536838475334], {10, + 24} -> Complex[ + 0.33333532926317844`, 0.0052698415932301845`], {11, 12} -> + Complex[0.4724699088969763, 0.], {11, 13} -> + Complex[ + 1.8041124150158796`*^-16, -1.457167719820518*^-16], {11, + 14} -> Complex[0.4724699088969763, 0.], {11, 15} -> + Complex[0.13800495411149877`, 0.3680867713233184], {11, + 16} -> Complex[-0.15558917647409173`, + 0.09979098803090053], {11, 17} -> + Complex[0.09710709374348331, -0.20541302638200343`], {11, + 18} -> Complex[-0.13183382282185513`, + 0.15967490110950291`], {11, 19} -> + Complex[0.13688251203715338`, -0.1036078618111689], {11, + 20} -> Complex[-0.3239368353423375, + 0.27853835234466173`], {11, 21} -> + Complex[0.25833102475550235`, -0.37839808713437206`], {11, + 22} -> Complex[ + 0.01476498538421066, -0.0004105289599095289], {11, 23} -> + Complex[-0.0868392972934217, 0.014513008624828502`], {11, + 24} -> Complex[-0.16585075463443222`, + 0.4085141858532183], {12, 13} -> + Complex[0.5275300911030236, 0.], {12, 14} -> + Complex[1.8041124150158796`*^-16, + 1.457167719820518*^-16], {12, 15} -> + Complex[-0.1905043599805688, 0.13762822610920328`], {12, + 16} -> Complex[0.1899783589314758, 0.1365082746823964], { + 12, 17} -> + Complex[-0.04249050665712762, 0.090268967382963], {12, 18} -> Complex[ - 0.06911881436797135, -0.04269376157510484], {15, 19} -> - Complex[0.0806509155236052, -0.06238441832735686], {15, - 20} -> Complex[-0.08181885113853848, - 0.0223150746441549], {15, 21} -> - Complex[0.3146513247958228, -0.026797378998685423`], {15, - 22} -> Complex[ - 0.011359633728347016`, -0.08520800008182422], {15, 23} -> - Complex[0.17837001887568327`, 0.09187489939370964], {15, - 24} -> Complex[-0.07164664525064873, - 0.2907134789193454], {16, 17} -> - Complex[0.1868664022204954, -0.06385028540854319], {16, - 18} -> Complex[-0.11463533571593523`, \ --0.055210078698593804`], {16, 19} -> - Complex[-0.2934854494126851, -0.04085502210736608], {16, + 0.11065480592987563`, 0.022498803837048064`], {12, 19} -> + Complex[0.017368709836144688`, -0.1497980533645717], {12, + 20} -> Complex[-0.19299966150123138`, \ +-0.3282026155242766], {12, 21} -> + Complex[-0.06250476454855013, 0.34969666557353624`], {12, + 22} -> Complex[-0.008146862343863803, + 0.10273781027166022`], {12, 23} -> + Complex[0.11666092373220538`, -0.218936695047531], {12, + 24} -> Complex[ + 0.161016937967343, 0.024863370726720582`], {13, 14} -> + Complex[0.4724699088969763, 0.], {13, 15} -> + Complex[0.13800495411149877`, 0.3680867713233184], {13, + 16} -> Complex[-0.15558917647409173`, + 0.09979098803090053], {13, 17} -> + Complex[0.09710709374348331, -0.20541302638200343`], {13, + 18} -> Complex[-0.13183382282185513`, + 0.15967490110950291`], {13, 19} -> + Complex[0.13688251203715338`, -0.1036078618111689], {13, + 20} -> Complex[-0.3239368353423375, + 0.27853835234466173`], {13, 21} -> + Complex[0.25833102475550235`, -0.37839808713437206`], {13, + 22} -> Complex[ + 0.01476498538421066, -0.0004105289599095289], {13, 23} -> + Complex[-0.0868392972934217, 0.014513008624828502`], {13, + 24} -> Complex[-0.16585075463443222`, + 0.4085141858532183], {14, 15} -> + Complex[-0.1905043599805688, 0.13762822610920328`], {14, + 16} -> Complex[0.1899783589314758, 0.1365082746823964], { + 14, 17} -> + Complex[-0.04249050665712762, 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Complex[-0.09727281618814543, -0.4251726947214757], {16, - 24} -> Complex[0.02023138254715448, -0.11137525641609], { - 17, 18} -> Complex[0.09630860197729411, 0.], {17, 19} -> - Complex[0.12702736104247997`, -0.01646149694625006], {17, - 20} -> Complex[-0.03764992141799037, - 0.13605099778794436`], {17, 21} -> - Complex[-0.07115780937991523, -0.011205895451017422`], { - 17, 22} -> - Complex[0.07416506719537758, 0.08051189278847616], {17, - 23} -> Complex[0.16774613988004144`, 0.225655277072902], { - 17, 24} -> - Complex[-0.09714752480970831, -0.05391368675976281], {18, - 19} -> Complex[-0.4024303768625533, \ --0.06319936655428628], {18, 20} -> - Complex[-0.02707098756043888, -0.08415309369657159], {18, - 21} -> Complex[0.6164020052722983, 0.03301684548082712], { - 18, 22} -> - Complex[-0.2385656438458264, -0.06903700176549488], {18, - 23} -> Complex[-0.24737746704016716`, \ --0.5022802985841684], {18, 24} -> - Complex[0.199408104684111, 0.11720650121625686`], {19, - 20} -> 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Complex[-0.11463533571593523`, -0.055210078698593804`], {10, 19} -> - Complex[-0.2934854494126851, -0.04085502210736608], {10, 20} -> - Complex[0.22168267144312054`, -0.03299920735716682], {10, 21} -> - Complex[-0.06237372995183321, 0.06727056435677253], {10, 22} -> - Complex[-0.030346695869449408`, -0.030149558835644692`], {10, 23} -> - Complex[-0.09727281618814543, -0.4251726947214757], {10, 24} -> - Complex[0.02023138254715448, -0.11137525641609], {11, 12} -> - Complex[0.5794387931418044, 0.], {11, 13} -> - Complex[-0.4009452410407828, -0.11284019268466222`], {11, 14} -> - Complex[0.5794387931418044, 0.], {11, 15} -> - Complex[-0.1634450723422762, 0.0011076658174966336`], {11, 16} -> - Complex[0.3689066714841379, 0.01963818646965646], {11, 17} -> - Complex[0.4351121808870284, 0.1282880779202612], {11, 18} -> - Complex[-0.06875971685231415, -0.006550328119388341], {11, 19} -> - Complex[-0.18425433538775948`, 0.03665899040288985], {11, 20} -> - Complex[-0.04214475287108173, -0.2125642839376449], {11, 21} -> - Complex[0.5691291778553188, 0.009546455336403484], {11, 22} -> - Complex[-0.11604982216622163`, -0.19926182212673743`], {11, 23} -> - Complex[-0.20476687273928368`, -0.20940927658482447`], {11, 24} -> - Complex[-0.04633762187626639, 0.37156097014999256`], {12, 13} -> - Complex[0.3744935743083003, 0.], {12, 14} -> - Complex[-0.4009452410407828, 0.11284019268466222`], {12, 15} -> - Complex[0.24390216997712094`, -0.12967162455970677`], {12, 16} -> - Complex[-0.364332505630415, 0.055629112414147225`], {12, 17} -> - Complex[-0.2300764357164763, -0.11137567802408056`], {12, 18} -> - Complex[-0.010533198723196371`, 0.006296053518233197], {12, 19} -> - Complex[-0.029186234486845222`, 0.022372160811117286`], {12, 20} -> - Complex[0.14142597329959572`, 0.061350391806736695`], {12, 21} -> - Complex[-0.33684082460892883`, 0.09005315780013691], {12, 22} -> - Complex[0.09059534447844704, 0.10345647613127328`], {12, 23} -> - Complex[-0.008851768247553363, -0.029068202106085572`], {12, 24} -> - Complex[-0.07795077274116374, -0.2904671042221048], {13, 14} -> - Complex[0.5794387931418044, 0.], {13, 15} -> - Complex[-0.1634450723422762, 0.0011076658174966336`], {13, 16} -> - Complex[0.3689066714841379, 0.01963818646965646], {13, 17} -> - Complex[0.4351121808870284, 0.1282880779202612], {13, 18} -> - Complex[-0.06875971685231415, -0.006550328119388341], {13, 19} -> - Complex[-0.18425433538775948`, 0.03665899040288985], {13, 20} -> - Complex[-0.04214475287108173, -0.2125642839376449], {13, 21} -> - Complex[0.5691291778553188, 0.009546455336403484], {13, 22} -> - Complex[-0.11604982216622163`, -0.19926182212673743`], {13, 23} -> - Complex[-0.20476687273928368`, -0.20940927658482447`], {13, 24} -> - Complex[-0.04633762187626639, 0.37156097014999256`], {14, 15} -> - Complex[0.24390216997712094`, -0.12967162455970677`], {14, 16} -> - Complex[-0.364332505630415, 0.055629112414147225`], {14, 17} -> - Complex[-0.2300764357164763, -0.11137567802408056`], {14, 18} -> - Complex[-0.010533198723196371`, 0.006296053518233197], {14, 19} -> - Complex[-0.029186234486845222`, 0.022372160811117286`], {14, 20} -> - Complex[0.14142597329959572`, 0.061350391806736695`], {14, 21} -> - Complex[-0.33684082460892883`, 0.09005315780013691], {14, 22} -> - Complex[0.09059534447844704, 0.10345647613127328`], {14, 23} -> - Complex[-0.008851768247553363, -0.029068202106085572`], {14, 24} -> - Complex[-0.07795077274116374, -0.2904671042221048], {15, 16} -> - Complex[0.4159087185308232, 0.], {15, 17} -> - Complex[0.14880448551485936`, 0.18343023977218464`], {15, 18} -> - Complex[0.06911881436797135, -0.04269376157510484], {15, 19} -> - Complex[0.0806509155236052, -0.06238441832735686], {15, 20} -> - Complex[-0.08181885113853848, 0.0223150746441549], {15, 21} -> - Complex[0.3146513247958228, -0.026797378998685423`], {15, 22} -> - Complex[0.011359633728347016`, -0.08520800008182422], {15, 23} -> - Complex[0.17837001887568327`, 0.09187489939370964], {15, 24} -> - Complex[-0.07164664525064873, 0.2907134789193454], {16, 17} -> - Complex[0.1868664022204954, -0.06385028540854319], {16, 18} -> - Complex[-0.11463533571593523`, -0.055210078698593804`], {16, 19} -> - Complex[-0.2934854494126851, -0.04085502210736608], {16, 20} -> - Complex[0.22168267144312054`, -0.03299920735716682], {16, 21} -> - Complex[-0.06237372995183321, 0.06727056435677253], {16, 22} -> - Complex[-0.030346695869449408`, -0.030149558835644692`], {16, 23} -> - Complex[-0.09727281618814543, -0.4251726947214757], {16, 24} -> - Complex[0.02023138254715448, -0.11137525641609], {17, 18} -> - Complex[0.09630860197729411, 0.], {17, 19} -> - Complex[0.12702736104247997`, -0.01646149694625006], {17, 20} -> - Complex[-0.03764992141799037, 0.13605099778794436`], {17, 21} -> - Complex[-0.07115780937991523, -0.011205895451017422`], {17, 22} -> - Complex[0.07416506719537758, 0.08051189278847616], {17, 23} -> - Complex[0.16774613988004144`, 0.225655277072902], {17, 24} -> - Complex[-0.09714752480970831, -0.05391368675976281], {18, 19} -> - Complex[-0.4024303768625533, -0.06319936655428628], {18, 20} -> - Complex[-0.02707098756043888, -0.08415309369657159], {18, 21} -> - Complex[0.6164020052722983, 0.03301684548082712], {18, 22} -> - Complex[-0.2385656438458264, -0.06903700176549488], {18, 23} -> - Complex[-0.24737746704016716`, -0.5022802985841684], {18, 24} -> - Complex[0.199408104684111, 0.11720650121625686`], {19, 20} -> - Complex[0.460087813322552, 0.], {19, 21} -> - Complex[-0.020878723063805726`, 0.1044611112500656], {19, 22} -> - Complex[0.24014852879671952`, -0.15329539227786965`], {19, 23} -> - Complex[0.11519091687891489`, -0.4012928958026726], {19, 24} -> - Complex[-0.45588071484204273`, 0.014752170920114424`], {20, 21} -> - Complex[-0.32198385415100245`, -0.056214448752933965`], {20, 22} -> - Complex[0.02087872306380592, 0.10446111125006569`], {20, 23} -> - Complex[0.34202714802636813`, 0.5203828231210746], {20, 24} -> - Complex[0.27130379014783124`, 0.12044715596755934`], {21, 22} -> - Complex[0.2677162255621613, 0.], {21, 23} -> - Complex[0.27244326195988916`, -0.011111878577649018`], {21, 24} -> - Complex[-0.3506829757791454, -0.10726001244941838`], {22, 23} -> - Complex[-0.29340373279675197`, -0.27809938962030384`], {22, 24} -> - Complex[-0.07968743240634073, 0.17702649633258472`]}, + Complex[-0.2782895722096578, 0.5061420791848172], {2, 4} -> + Complex[-0.06947320790243908, -0.12537107773431694`], {2, 5} -> + Complex[-0.06797212261961828, -0.14596193524618625`], {2, 6} -> + Complex[0.029019428968993538`, -0.21528502386671927`], {2, 7} -> + Complex[-0.16418162796222252`, -0.13133580222113328`], {2, 8} -> + Complex[-0.23097227051538433`, 0.42320695740719316`], {2, 9} -> + Complex[-0.2917595352077467, -0.3784536838475334], {2, 10} -> + Complex[0.01747597846754094, -0.10203270358947614`], {2, 11} -> + Complex[0.11666092373220538`, 0.218936695047531], {2, 12} -> + Complex[-0.0868392972934217, -0.014513008624828502`], {2, 13} -> + Complex[0.11666092373220538`, 0.218936695047531], {2, 14} -> + Complex[-0.0868392972934217, -0.014513008624828502`], {2, 15} -> + Complex[-0.2917595352077467, -0.3784536838475334], {2, 16} -> + Complex[0.01747597846754094, -0.10203270358947614`], {2, 17} -> + Complex[-0.16418162796222252`, -0.13133580222113328`], {2, 18} -> + Complex[-0.23097227051538433`, 0.42320695740719316`], {2, 19} -> + Complex[-0.06797212261961828, -0.14596193524618625`], {2, 20} -> + Complex[0.029019428968993538`, -0.21528502386671927`], {2, 21} -> + Complex[-0.2782895722096578, 0.5061420791848172], {2, 22} -> + Complex[-0.06947320790243908, -0.12537107773431694`], {2, 23} -> + 1, {3, 4} -> Complex[0.12373001666450631`, 0.], {3, 5} -> + Complex[-0.029720518023860276`, 0.053544088868121745`], {3, 6} -> + Complex[-0.15211930857597808`, 0.0462428201376506], {3, 7} -> + Complex[0.053161902759652525`, 0.14451247653181823`], {3, 8} -> + Complex[-0.13167457410588743`, 0.003054508743256395], {3, 9} -> + Complex[0.08210404404737587, -0.05307307167276523], {3, 10} -> + Complex[-0.04368290380765367, 0.027436459786564067`], {3, 11} -> + Complex[-0.008146862343863803, -0.10273781027166022`], {3, 12} -> + Complex[0.01476498538421066, 0.0004105289599095289], {3, 13} -> + Complex[-0.008146862343863803, -0.10273781027166022`], {3, 14} -> + Complex[0.01476498538421066, 0.0004105289599095289], {3, 15} -> + Complex[0.08210404404737587, -0.05307307167276523], {3, 16} -> + Complex[-0.04368290380765367, 0.027436459786564067`], {3, 17} -> + Complex[0.053161902759652525`, 0.14451247653181823`], {3, 18} -> + Complex[-0.13167457410588743`, 0.003054508743256395], {3, 19} -> + Complex[-0.029720518023860276`, 0.053544088868121745`], {3, 20} -> + Complex[-0.15211930857597808`, 0.0462428201376506], {3, 21} -> + Complex[-1.1102230246251565`*^-16, 0.], {3, 22} -> + Complex[0.12373001666450631`, 0.], {3, 23} -> + Complex[-0.06947320790243908, 0.12537107773431694`], {3, 24} -> + Complex[-0.1766768302665536, -0.1352957631030186], {4, 5} -> + Complex[0.030636763457055183`, 0.11355874224143539`], {4, 6} -> + Complex[-0.5748910514001031, 0.14633045383134305`], {4, 7} -> + Complex[0.2616855917631509, 0.009667937264448562], {4, 8} -> + Complex[0.037739555243020126`, 0.012308050579545225`], {4, 9} -> + Complex[-0.2346066684020043, 0.5335785389713812], {4, 10} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {4, 11} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {4, 12} -> + Complex[0.25833102475550235`, 0.37839808713437206`], {4, 13} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {4, 14} -> + Complex[0.25833102475550235`, 0.37839808713437206`], {4, 15} -> + Complex[-0.2346066684020043, 0.5335785389713812], {4, 16} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {4, 17} -> + Complex[0.2616855917631509, 0.009667937264448562], {4, 18} -> + Complex[0.037739555243020126`, 0.012308050579545225`], {4, 19} -> + Complex[0.030636763457055183`, 0.11355874224143539`], {4, 20} -> + Complex[-0.5748910514001031, 0.14633045383134305`], {4, 21} -> + Complex[0.8762699833354937, 0.], {4, 22} -> + Complex[-1.1102230246251565`*^-16, 0.], {4, 23} -> + Complex[-0.2782895722096578, -0.5061420791848172], {4, 24} -> + Complex[-0.41976082253431385`, 0.08596477743313696], {5, 6} -> + Complex[0.7835951260215941, 0.], {5, 7} -> + Complex[-0.17834040611966948`, -0.12609948481270494`], {5, 8} -> + Complex[0.2578656808252187, -0.10876030801336257`], {5, 9} -> + Complex[0.12289396168675226`, -0.38209796458606343`], {5, 10} -> + Complex[0.13476314496413244`, 0.0767423516894553], {5, 11} -> + Complex[-0.19299966150123138`, 0.3282026155242766], {5, 12} -> + Complex[-0.3239368353423375, -0.27853835234466173`], {5, 13} -> + Complex[-0.19299966150123138`, 0.3282026155242766], {5, 14} -> + Complex[-0.3239368353423375, -0.27853835234466173`], {5, 15} -> + Complex[0.12289396168675226`, -0.38209796458606343`], {5, 16} -> + Complex[0.13476314496413244`, 0.0767423516894553], {5, 17} -> + Complex[-0.17834040611966948`, -0.12609948481270494`], {5, 18} -> + Complex[0.2578656808252187, -0.10876030801336257`], {5, 19} -> + Complex[8.326672684688674*^-17, -1.3877787807814457`*^-17], {5, + 20} -> Complex[0.7835951260215941, 0.], {5, 21} -> + Complex[-0.5748910514001031, -0.14633045383134305`], {5, 22} -> + Complex[-0.15211930857597808`, -0.0462428201376506], {5, 23} -> + Complex[0.029019428968993538`, 0.21528502386671927`], {5, 24} -> + Complex[0.46472596940822686`, 0.06199143840307167], {6, 7} -> + Complex[0.22516518597054483`, -0.1013577129442031], {6, 8} -> + Complex[-0.007671097858546656, -0.0057340892009139326`], {6, 9} -> + Complex[-0.09460977915945387, 0.11933166053315665`], {6, 10} -> + Complex[0.02081012384838236, 0.14234469229568147`], {6, 11} -> + Complex[0.017368709836144688`, 0.1497980533645717], {6, 12} -> + Complex[0.13688251203715338`, 0.1036078618111689], {6, 13} -> + Complex[0.017368709836144688`, 0.1497980533645717], {6, 14} -> + Complex[0.13688251203715338`, 0.1036078618111689], {6, 15} -> + Complex[-0.09460977915945387, 0.11933166053315665`], {6, 16} -> + Complex[0.02081012384838236, 0.14234469229568147`], {6, 17} -> + Complex[0.22516518597054483`, -0.1013577129442031], {6, 18} -> + Complex[-0.007671097858546656, -0.0057340892009139326`], {6, 19} -> + Complex[0.21640487397840627`, 0.], {6, 20} -> + Complex[8.326672684688674*^-17, 1.3877787807814457`*^-17], {6, 21} -> + Complex[0.030636763457055183`, -0.11355874224143539`], {6, 22} -> + Complex[-0.029720518023860276`, -0.053544088868121745`], {6, 23} -> + Complex[-0.06797212261961828, 0.14596193524618625`], {6, 24} -> + Complex[-0.1599463611986579, 0.3186964622408383], {7, 8} -> + Complex[0.5226137655938699, 0.], {7, 9} -> + Complex[-0.08485061223426603, 0.11980477813807351`], {7, 10} -> + Complex[0.17479296155780188`, 0.14732776197358494`], {7, 11} -> + Complex[0.11065480592987563`, 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RowBox[{"0.10726001244941838`", " ", "\[ImaginaryI]"}]}], + RowBox[{"-", "0.41976082253431385`"}], "-", + RowBox[{"0.08596477743313696`", " ", "\[ImaginaryI]"}]}], ContentPadding -> False, FrameMargins -> 0, StripOnInput -> True, BaselinePosition -> Baseline, ImageSize -> {{1, 300}, Automatic}], LineBreakWithin -> @@ -19146,8 +19352,8 @@ T/JX6cbTzYduvXT7Sfe8SOdOwtX/AIomsy8= StyleBox[ PaneBox[ RowBox[{ - RowBox[{"0.27130379014783124`", "\[VeryThinSpace]"}], "-", - RowBox[{"0.12044715596755934`", " ", "\[ImaginaryI]"}]}], + RowBox[{"0.46472596940822686`", "\[VeryThinSpace]"}], "-", + RowBox[{"0.06199143840307167`", " ", "\[ImaginaryI]"}]}], ContentPadding -> False, FrameMargins -> 0, StripOnInput -> True, BaselinePosition -> Baseline, ImageSize -> {{1, 300}, Automatic}], LineBreakWithin -> @@ -19158,9 +19364,9 @@ T/JX6cbTzYduvXT7Sfe8SOdOwtX/AIomsy8= StyleBox[ PaneBox[ RowBox[{ - RowBox[{"-", "0.45588071484204273`"}], "-", - RowBox[{"0.014752170920114424`", " ", "\[ImaginaryI]"}]}], - ContentPadding -> False, FrameMargins -> 0, StripOnInput -> + RowBox[{"-", "0.1599463611986579`"}], "-", + RowBox[{"0.3186964622408383`", " ", "\[ImaginaryI]"}]}], + ContentPadding -> False, FrameMargins -> 0, StripOnInput -> True, BaselinePosition -> Baseline, ImageSize -> {{1, 300}, Automatic}], LineBreakWithin -> False]}]}, { @@ -19191,391 +19397,395 @@ T/JX6cbTzYduvXT7Sfe8SOdOwtX/AIomsy8= StructuredArray`StructuredData[{24, 24}, {{{1, 3} -> - Complex[-0.07968743240634073, -0.17702649633258472`], {1, - 4} -> Complex[-0.3506829757791454, - 0.10726001244941838`], {1, 5} -> - Complex[0.27130379014783124`, -0.12044715596755934`], {1, - 6} -> Complex[-0.45588071484204273`, \ --0.014752170920114424`], {1, 7} -> - Complex[0.199408104684111, -0.11720650121625686`], {1, 8} -> - Complex[-0.09714752480970831, 0.05391368675976281], {1, - 9} -> Complex[-0.07164664525064873, -0.2907134789193454], \ -{1, 10} -> Complex[0.02023138254715448, 0.11137525641609], {1, 11} -> - Complex[-0.07795077274116374, 0.2904671042221048], {1, - 12} -> Complex[-0.04633762187626639, \ --0.37156097014999256`], {1, 13} -> - Complex[-0.07795077274116374, 0.2904671042221048], {1, - 14} -> Complex[-0.04633762187626639, \ --0.37156097014999256`], {1, 15} -> - Complex[-0.07164664525064873, -0.2907134789193454], {1, - 16} -> Complex[0.02023138254715448, 0.11137525641609], {1, - 17} -> Complex[ - 0.199408104684111, -0.11720650121625686`], {1, 18} -> - Complex[-0.09714752480970831, 0.05391368675976281], {1, - 19} -> Complex[ - 0.27130379014783124`, -0.12044715596755934`], {1, 20} -> - Complex[-0.45588071484204273`, -0.014752170920114424`], { - 1, 21} -> - Complex[-0.07968743240634073, -0.17702649633258472`], {1, - 22} -> Complex[-0.3506829757791454, - 0.10726001244941838`], {1, 24} -> 1, {2, 3} -> - Complex[-0.29340373279675197`, 0.27809938962030384`], {2, - 4} -> Complex[ - 0.27244326195988916`, 0.011111878577649018`], {2, 5} -> - Complex[0.34202714802636813`, -0.5203828231210746], {2, - 6} -> Complex[0.11519091687891489`, 0.4012928958026726], { - 2, 7} -> Complex[-0.24737746704016716`, - 0.5022802985841684], {2, 8} -> - Complex[0.16774613988004144`, -0.225655277072902], {2, 9} -> - Complex[0.17837001887568327`, -0.09187489939370964], {2, - 10} -> Complex[-0.09727281618814543, - 0.4251726947214757], {2, 11} -> - Complex[-0.008851768247553363, 0.029068202106085572`], {2, - 12} -> Complex[-0.20476687273928368`, - 0.20940927658482447`], {2, 13} -> - Complex[-0.008851768247553363, 0.029068202106085572`], {2, - 14} -> Complex[-0.20476687273928368`, - 0.20940927658482447`], {2, 15} -> - Complex[0.17837001887568327`, -0.09187489939370964], {2, - 16} -> Complex[-0.09727281618814543, - 0.4251726947214757], {2, 17} -> - Complex[-0.24737746704016716`, 0.5022802985841684], {2, - 18} -> Complex[ - 0.16774613988004144`, -0.225655277072902], {2, 19} -> - Complex[0.34202714802636813`, -0.5203828231210746], {2, - 20} -> Complex[ - 0.11519091687891489`, 0.4012928958026726], {2, 21} -> - Complex[-0.29340373279675197`, 0.27809938962030384`], {2, - 22} -> Complex[ - 0.27244326195988916`, 0.011111878577649018`], {2, 23} -> - 1, {3, 4} -> Complex[0.2677162255621613, 0.], {3, 5} -> - Complex[0.02087872306380592, -0.10446111125006569`], {3, + Complex[-0.1766768302665536, 0.1352957631030186], {1, 4} -> + Complex[-0.41976082253431385`, -0.08596477743313696], {1, + 5} -> Complex[ + 0.46472596940822686`, -0.06199143840307167], {1, 6} -> + Complex[-0.1599463611986579, -0.3186964622408383], {1, 7} -> + Complex[-0.42262163352679005`, -0.31456841745062747`], {1, + 8} -> Complex[ + 0.29687889692943875`, -0.2375326567068352], {1, 9} -> + Complex[0.33333532926317844`, -0.0052698415932301845`], { + 1, 10} -> Complex[ + 0.34087862056969404`, -0.1100388283919505], {1, 11} -> + Complex[-0.16585075463443222`, -0.4085141858532183], {1, + 12} -> Complex[ + 0.161016937967343, -0.024863370726720582`], {1, 13} -> + Complex[-0.16585075463443222`, -0.4085141858532183], {1, + 14} -> Complex[ + 0.161016937967343, -0.024863370726720582`], {1, 15} -> + Complex[0.33333532926317844`, -0.0052698415932301845`], { + 1, 16} -> Complex[ + 0.34087862056969404`, -0.1100388283919505], {1, 17} -> + Complex[-0.42262163352679005`, -0.31456841745062747`], {1, + 18} -> Complex[ + 0.29687889692943875`, -0.2375326567068352], {1, 19} -> + Complex[0.46472596940822686`, -0.06199143840307167], {1, + 20} -> Complex[-0.1599463611986579, -0.3186964622408383], \ +{1, 21} -> Complex[-0.1766768302665536, 0.1352957631030186], {1, 22} -> + Complex[-0.41976082253431385`, -0.08596477743313696], {1, + 24} -> 1, {2, 3} -> + Complex[-0.06947320790243908, -0.12537107773431694`], {2, + 4} -> Complex[-0.2782895722096578, 0.5061420791848172], { + 2, 5} -> Complex[ + 0.029019428968993538`, -0.21528502386671927`], {2, 6} -> + Complex[-0.06797212261961828, -0.14596193524618625`], {2, + 7} -> Complex[-0.16418162796222252`, \ +-0.13133580222113328`], {2, 8} -> + Complex[-0.23097227051538433`, 0.42320695740719316`], {2, + 9} -> Complex[-0.2917595352077467, -0.3784536838475334], { + 2, 10} -> Complex[ + 0.01747597846754094, -0.10203270358947614`], {2, 11} -> + Complex[-0.0868392972934217, -0.014513008624828502`], {2, + 12} -> Complex[0.11666092373220538`, 0.218936695047531], { + 2, 13} -> + Complex[-0.0868392972934217, -0.014513008624828502`], {2, + 14} -> Complex[0.11666092373220538`, 0.218936695047531], { + 2, 15} -> + Complex[-0.2917595352077467, -0.3784536838475334], {2, + 16} -> Complex[ + 0.01747597846754094, -0.10203270358947614`], {2, 17} -> + Complex[-0.16418162796222252`, -0.13133580222113328`], {2, + 18} -> Complex[-0.23097227051538433`, + 0.42320695740719316`], {2, 19} -> + Complex[0.029019428968993538`, -0.21528502386671927`], {2, + 20} -> + Complex[-0.06797212261961828, -0.14596193524618625`], {2, + 21} -> Complex[-0.06947320790243908, \ +-0.12537107773431694`], {2, 22} -> + Complex[-0.2782895722096578, 0.5061420791848172], {2, 23} -> + 1, {3, 4} -> Complex[0.8762699833354937, 0.], {3, 5} -> + Complex[-0.5748910514001031, 0.14633045383134305`], {3, 6} -> Complex[ - 0.24014852879671952`, 0.15329539227786965`], {3, 7} -> - Complex[-0.2385656438458264, 0.06903700176549488], {3, 8} -> - Complex[0.07416506719537758, -0.08051189278847616], {3, - 9} -> Complex[ - 0.011359633728347016`, 0.08520800008182422], {3, 10} -> - Complex[-0.030346695869449408`, 0.030149558835644692`], { - 3, 11} -> Complex[ - 0.09059534447844704, -0.10345647613127328`], {3, 12} -> - Complex[-0.11604982216622163`, 0.19926182212673743`], {3, + 0.030636763457055183`, 0.11355874224143539`], {3, 7} -> + Complex[0.2616855917631509, 0.009667937264448562], {3, 8} -> + Complex[0.037739555243020126`, 0.012308050579545225`], {3, + 9} -> Complex[-0.2346066684020043, 0.5335785389713812], { + 3, 10} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {3, + 11} -> Complex[ + 0.25833102475550235`, 0.37839808713437206`], {3, 12} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {3, 13} -> Complex[ - 0.09059534447844704, -0.10345647613127328`], {3, 14} -> - Complex[-0.11604982216622163`, 0.19926182212673743`], {3, - 15} -> Complex[ - 0.011359633728347016`, 0.08520800008182422], {3, 16} -> - Complex[-0.030346695869449408`, 0.030149558835644692`], { - 3, 17} -> Complex[-0.2385656438458264, - 0.06903700176549488], {3, 18} -> - Complex[0.07416506719537758, -0.08051189278847616], {3, - 19} -> Complex[ - 0.02087872306380592, -0.10446111125006569`], {3, 20} -> - Complex[0.24014852879671952`, 0.15329539227786965`], {3, - 21} -> Complex[-0.1157253008903192, - 0.07182611485724257], {3, 22} -> - Complex[0.2677162255621613, 0.], {3, 23} -> - Complex[0.27244326195988916`, -0.011111878577649018`], {3, - 24} -> - Complex[-0.3506829757791454, -0.10726001244941838`], {4, - 5} -> Complex[-0.32198385415100245`, - 0.056214448752933965`], {4, 6} -> - Complex[-0.020878723063805726`, -0.1044611112500656], {4, - 7} -> Complex[0.6164020052722983, -0.03301684548082712], { - 4, 8} -> Complex[-0.07115780937991523, - 0.011205895451017422`], {4, 9} -> - Complex[0.3146513247958228, 0.026797378998685423`], {4, - 10} -> Complex[-0.06237372995183321, \ --0.06727056435677253], {4, 11} -> - Complex[-0.33684082460892883`, -0.09005315780013691], {4, + 0.25833102475550235`, 0.37839808713437206`], {3, 14} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {3, + 15} -> Complex[-0.2346066684020043, 0.5335785389713812], { + 3, 16} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {3, + 17} -> Complex[ + 0.2616855917631509, 0.009667937264448562], {3, 18} -> + Complex[0.037739555243020126`, 0.012308050579545225`], {3, + 19} -> Complex[-0.5748910514001031, + 0.14633045383134305`], {3, 20} -> + Complex[0.030636763457055183`, 0.11355874224143539`], {3, + 21} -> Complex[-1.1102230246251565`*^-16, 0.], {3, 22} -> + Complex[0.8762699833354937, 0.], {3, 23} -> + Complex[-0.2782895722096578, -0.5061420791848172], {3, + 24} -> Complex[-0.41976082253431385`, + 0.08596477743313696], {4, 5} -> + Complex[-0.15211930857597808`, 0.0462428201376506], {4, + 6} -> Complex[-0.029720518023860276`, + 0.053544088868121745`], {4, 7} -> + Complex[0.053161902759652525`, 0.14451247653181823`], {4, + 8} -> Complex[-0.13167457410588743`, + 0.003054508743256395], {4, 9} -> + Complex[0.08210404404737587, -0.05307307167276523], {4, + 10} -> Complex[-0.04368290380765367, + 0.027436459786564067`], {4, 11} -> + Complex[0.01476498538421066, 0.0004105289599095289], {4, + 12} -> Complex[-0.008146862343863803, \ +-0.10273781027166022`], {4, 13} -> + Complex[0.01476498538421066, 0.0004105289599095289], {4, + 14} -> Complex[-0.008146862343863803, \ +-0.10273781027166022`], {4, 15} -> + Complex[0.08210404404737587, -0.05307307167276523], {4, + 16} -> Complex[-0.04368290380765367, + 0.027436459786564067`], {4, 17} -> + Complex[0.053161902759652525`, 0.14451247653181823`], {4, + 18} -> Complex[-0.13167457410588743`, + 0.003054508743256395], {4, 19} -> + Complex[-0.15211930857597808`, 0.0462428201376506], {4, + 20} -> Complex[-0.029720518023860276`, + 0.053544088868121745`], {4, 21} -> + Complex[0.12373001666450631`, 0.], {4, 22} -> + Complex[-1.1102230246251565`*^-16, 0.], {4, 23} -> + Complex[-0.06947320790243908, 0.12537107773431694`], {4, + 24} -> Complex[-0.1766768302665536, -0.1352957631030186], \ +{5, 6} -> Complex[0.21640487397840627`, 0.], {5, 7} -> + Complex[0.22516518597054483`, -0.1013577129442031], {5, + 8} -> Complex[-0.007671097858546656, \ +-0.0057340892009139326`], {5, 9} -> + Complex[-0.09460977915945387, 0.11933166053315665`], {5, + 10} -> Complex[ + 0.02081012384838236, 0.14234469229568147`], {5, 11} -> + Complex[0.13688251203715338`, 0.1036078618111689], {5, 12} -> Complex[ - 0.5691291778553188, -0.009546455336403484], {4, 13} -> - Complex[-0.33684082460892883`, -0.09005315780013691], {4, + 0.017368709836144688`, 0.1497980533645717], {5, 13} -> + Complex[0.13688251203715338`, 0.1036078618111689], {5, 14} -> Complex[ - 0.5691291778553188, -0.009546455336403484], {4, 15} -> - Complex[0.3146513247958228, 0.026797378998685423`], {4, - 16} -> Complex[-0.06237372995183321, \ --0.06727056435677253], {4, 17} -> - Complex[0.6164020052722983, -0.03301684548082712], {4, - 18} -> Complex[-0.07115780937991523, - 0.011205895451017422`], {4, 19} -> - Complex[-0.32198385415100245`, 0.056214448752933965`], {4, - 20} -> - Complex[-0.020878723063805726`, -0.1044611112500656], {4, - 21} -> Complex[0.7322837744378387, 0.], {4, 22} -> - Complex[-0.1157253008903192, -0.07182611485724257], {4, - 23} -> Complex[-0.29340373279675197`, \ --0.27809938962030384`], {4, 24} -> - Complex[-0.07968743240634073, 0.17702649633258472`], {5, - 6} -> Complex[0.460087813322552, 0.], {5, 7} -> - Complex[-0.02707098756043888, 0.08415309369657159], {5, - 8} -> Complex[-0.03764992141799037, \ --0.13605099778794436`], {5, 9} -> - Complex[-0.08181885113853848, -0.0223150746441549], {5, + 0.017368709836144688`, 0.1497980533645717], {5, 15} -> + Complex[-0.09460977915945387, 0.11933166053315665`], {5, + 16} -> Complex[ + 0.02081012384838236, 0.14234469229568147`], {5, 17} -> + Complex[0.22516518597054483`, -0.1013577129442031], {5, + 18} -> Complex[-0.007671097858546656, \ +-0.0057340892009139326`], {5, 19} -> + Complex[8.326672684688674*^-17, + 1.3877787807814457`*^-17], {5, 20} -> + Complex[0.21640487397840627`, 0.], {5, 21} -> + Complex[-0.029720518023860276`, -0.053544088868121745`], { + 5, 22} -> Complex[ + 0.030636763457055183`, -0.11355874224143539`], {5, 23} -> + Complex[-0.06797212261961828, 0.14596193524618625`], {5, + 24} -> Complex[-0.1599463611986579, 0.3186964622408383], { + 6, 7} -> + Complex[-0.17834040611966948`, -0.12609948481270494`], {6, + 8} -> Complex[ + 0.2578656808252187, -0.10876030801336257`], {6, 9} -> + Complex[0.12289396168675226`, -0.38209796458606343`], {6, 10} -> Complex[ - 0.22168267144312054`, 0.03299920735716682], {5, 11} -> - Complex[0.14142597329959572`, -0.061350391806736695`], {5, - 12} -> Complex[-0.04214475287108173, - 0.2125642839376449], {5, 13} -> - Complex[0.14142597329959572`, -0.061350391806736695`], {5, - 14} -> Complex[-0.04214475287108173, - 0.2125642839376449], {5, 15} -> - Complex[-0.08181885113853848, -0.0223150746441549], {5, + 0.13476314496413244`, 0.0767423516894553], {6, 11} -> + Complex[-0.3239368353423375, -0.27853835234466173`], {6, + 12} -> Complex[-0.19299966150123138`, + 0.3282026155242766], {6, 13} -> + Complex[-0.3239368353423375, -0.27853835234466173`], {6, + 14} -> Complex[-0.19299966150123138`, + 0.3282026155242766], {6, 15} -> + Complex[0.12289396168675226`, -0.38209796458606343`], {6, 16} -> Complex[ - 0.22168267144312054`, 0.03299920735716682], {5, 17} -> - Complex[-0.02707098756043888, 0.08415309369657159], {5, - 18} -> Complex[-0.03764992141799037, \ --0.13605099778794436`], {5, 19} -> - Complex[-0.27467400107696954`, -0.08314412135251599], {5, - 20} -> Complex[0.460087813322552, 0.], {5, 21} -> - Complex[-0.020878723063805726`, 0.1044611112500656], {5, - 22} -> Complex[ - 0.24014852879671952`, -0.15329539227786965`], {5, 23} -> - Complex[0.11519091687891489`, -0.4012928958026726], {5, - 24} -> Complex[-0.45588071484204273`, - 0.014752170920114424`], {6, 7} -> - Complex[-0.4024303768625533, 0.06319936655428628], {6, 8} -> - Complex[0.12702736104247997`, 0.01646149694625006], {6, - 9} -> Complex[0.0806509155236052, 0.06238441832735686], { - 6, 10} -> Complex[-0.2934854494126851, - 0.04085502210736608], {6, 11} -> - Complex[-0.029186234486845222`, -0.022372160811117286`], { - 6, 12} -> - Complex[-0.18425433538775948`, -0.03665899040288985], {6, - 13} -> Complex[-0.029186234486845222`, \ --0.022372160811117286`], {6, 14} -> - Complex[-0.18425433538775948`, -0.03665899040288985], {6, - 15} -> Complex[0.0806509155236052, 0.06238441832735686], { - 6, 16} -> Complex[-0.2934854494126851, - 0.04085502210736608], {6, 17} -> - Complex[-0.4024303768625533, 0.06319936655428628], {6, - 18} -> Complex[ - 0.12702736104247997`, 0.01646149694625006], {6, 19} -> - Complex[0.5399121866774477, 0.], {6, 20} -> - Complex[-0.27467400107696954`, 0.08314412135251599], {6, - 21} -> Complex[-0.32198385415100245`, \ --0.056214448752933965`], {6, 22} -> - Complex[0.02087872306380592, 0.10446111125006569`], {6, + 0.13476314496413244`, 0.0767423516894553], {6, 17} -> + Complex[-0.17834040611966948`, -0.12609948481270494`], {6, + 18} -> Complex[ + 0.2578656808252187, -0.10876030801336257`], {6, 19} -> + Complex[0.7835951260215941, 0.], {6, 20} -> + Complex[ + 8.326672684688674*^-17, -1.3877787807814457`*^-17], {6, + 21} -> Complex[-0.15211930857597808`, \ +-0.0462428201376506], {6, 22} -> + Complex[-0.5748910514001031, -0.14633045383134305`], {6, 23} -> Complex[ - 0.34202714802636813`, 0.5203828231210746], {6, 24} -> - Complex[0.27130379014783124`, 0.12044715596755934`], {7, - 8} -> Complex[0.09630860197729411, 0.], {7, 9} -> - Complex[0.06911881436797135, 0.04269376157510484], {7, - 10} -> Complex[-0.11463533571593523`, - 0.055210078698593804`], {7, 11} -> - Complex[-0.010533198723196371`, -0.006296053518233197], { - 7, 12} -> Complex[-0.06875971685231415, - 0.006550328119388341], {7, 13} -> - Complex[-0.010533198723196371`, -0.006296053518233197], { - 7, 14} -> Complex[-0.06875971685231415, - 0.006550328119388341], {7, 15} -> - Complex[0.06911881436797135, 0.04269376157510484], {7, - 16} -> Complex[-0.11463533571593523`, - 0.055210078698593804`], {7, 17} -> - Complex[-0.13812244808851923`, 0.022678210677030422`], {7, - 18} -> Complex[0.09630860197729411, 0.], {7, 19} -> - Complex[0.12702736104247997`, -0.01646149694625006], {7, - 20} -> Complex[-0.03764992141799037, - 0.13605099778794436`], {7, 21} -> - Complex[-0.07115780937991523, -0.011205895451017422`], {7, + 0.029019428968993538`, 0.21528502386671927`], {6, 24} -> + Complex[0.46472596940822686`, 0.06199143840307167], {7, + 8} -> Complex[0.5226137655938699, 0.], {7, 9} -> + Complex[-0.08485061223426603, 0.11980477813807351`], {7, + 10} -> Complex[ + 0.17479296155780188`, 0.14732776197358494`], {7, 11} -> + Complex[-0.13183382282185513`, -0.15967490110950291`], {7, + 12} -> Complex[ + 0.11065480592987563`, -0.022498803837048064`], {7, 13} -> + Complex[-0.13183382282185513`, -0.15967490110950291`], {7, + 14} -> Complex[ + 0.11065480592987563`, -0.022498803837048064`], {7, 15} -> + Complex[-0.08485061223426603, 0.11980477813807351`], {7, + 16} -> Complex[ + 0.17479296155780188`, 0.14732776197358494`], {7, 17} -> + Complex[-1.7867651802561116`*^-16, + 1.1449174941446927`*^-16], {7, 18} -> + Complex[0.5226137655938699, 0.], {7, 19} -> + Complex[0.2578656808252187, 0.10876030801336257`], {7, + 20} -> Complex[-0.007671097858546656, + 0.0057340892009139326`], {7, 21} -> + Complex[-0.13167457410588743`, -0.003054508743256395], {7, 22} -> Complex[ - 0.07416506719537758, 0.08051189278847616], {7, 23} -> - Complex[0.16774613988004144`, 0.225655277072902], {7, 24} -> - Complex[-0.09714752480970831, -0.05391368675976281], {8, - 9} -> Complex[ - 0.14880448551485936`, -0.18343023977218464`], {8, 10} -> - Complex[0.1868664022204954, 0.06385028540854319], {8, 11} -> - Complex[-0.2300764357164763, 0.11137567802408056`], {8, - 12} -> Complex[0.4351121808870284, -0.1282880779202612], { - 8, 13} -> Complex[-0.2300764357164763, - 0.11137567802408056`], {8, 14} -> - Complex[0.4351121808870284, -0.1282880779202612], {8, 15} -> - Complex[0.14880448551485936`, -0.18343023977218464`], {8, - 16} -> Complex[0.1868664022204954, 0.06385028540854319], { - 8, 17} -> Complex[0.7694387170582185, 0.], {8, 18} -> - Complex[-0.13812244808851923`, -0.022678210677030422`], { - 8, 19} -> - Complex[-0.4024303768625533, -0.06319936655428628], {8, - 20} -> Complex[-0.02707098756043888, \ --0.08415309369657159], {8, 21} -> - Complex[0.6164020052722983, 0.03301684548082712], {8, 22} -> - Complex[-0.2385656438458264, -0.06903700176549488], {8, - 23} -> Complex[-0.24737746704016716`, \ --0.5022802985841684], {8, 24} -> - Complex[0.199408104684111, 0.11720650121625686`], {9, 10} -> - Complex[0.41416215192864014`, 0.], {9, 11} -> - Complex[0.24390216997712094`, 0.12967162455970677`], {9, - 12} -> Complex[-0.1634450723422762, \ --0.0011076658174966336`], {9, 13} -> - Complex[0.24390216997712094`, 0.12967162455970677`], {9, - 14} -> Complex[-0.1634450723422762, \ --0.0011076658174966336`], {9, 15} -> - Complex[-0.29089675324323006`, -0.1558046632134409], {9, - 16} -> Complex[0.41416215192864014`, 0.], {9, 17} -> - Complex[0.1868664022204954, -0.06385028540854319], {9, - 18} -> Complex[-0.11463533571593523`, \ --0.055210078698593804`], {9, 19} -> - Complex[-0.2934854494126851, -0.04085502210736608], {9, + 0.037739555243020126`, -0.012308050579545225`], {7, 23} -> + Complex[-0.23097227051538433`, -0.42320695740719316`], {7, + 24} -> Complex[ + 0.29687889692943875`, 0.2375326567068352], {8, 9} -> + Complex[-0.17479296155780177`, 0.14732776197358502`], {8, + 10} -> Complex[ + 0.08485061223426596, 0.11980477813807357`], {8, 11} -> + Complex[0.09710709374348331, 0.20541302638200343`], {8, + 12} -> Complex[-0.04249050665712762, -0.090268967382963], \ +{8, 13} -> Complex[0.09710709374348331, 0.20541302638200343`], {8, 14} -> + Complex[-0.04249050665712762, -0.090268967382963], {8, + 15} -> Complex[-0.17479296155780177`, + 0.14732776197358502`], {8, 16} -> + Complex[0.08485061223426596, 0.11980477813807357`], {8, + 17} -> Complex[0.47738623440613004`, 0.], {8, 18} -> + Complex[-1.7867651802561116`*^-16, \ +-1.1449174941446927`*^-16], {8, 19} -> + Complex[-0.17834040611966948`, 0.12609948481270494`], {8, 20} -> Complex[ - 0.22168267144312054`, -0.03299920735716682], {9, 21} -> - Complex[-0.06237372995183321, 0.06727056435677253], {9, - 22} -> Complex[-0.030346695869449408`, \ --0.030149558835644692`], {9, 23} -> - Complex[-0.09727281618814543, -0.4251726947214757], {9, - 24} -> Complex[0.02023138254715448, -0.11137525641609], { - 10, 11} -> - Complex[-0.364332505630415, -0.055629112414147225`], {10, - 12} -> Complex[ - 0.3689066714841379, -0.01963818646965646], {10, 13} -> - Complex[-0.364332505630415, -0.055629112414147225`], {10, - 14} -> Complex[ - 0.3689066714841379, -0.01963818646965646], {10, 15} -> - Complex[0.4159087185308232, 0.], {10, 16} -> - Complex[-0.29089675324323006`, 0.1558046632134409], {10, + 0.22516518597054483`, 0.1013577129442031], {8, 21} -> + Complex[0.053161902759652525`, -0.14451247653181823`], {8, + 22} -> Complex[ + 0.2616855917631509, -0.009667937264448562], {8, 23} -> + Complex[-0.16418162796222252`, 0.13133580222113328`], {8, + 24} -> Complex[-0.42262163352679005`, + 0.31456841745062747`], {9, 10} -> + Complex[0.3800250972969439, 0.], {9, 11} -> + Complex[-0.15558917647409173`, -0.09979098803090053], {9, + 12} -> Complex[0.1899783589314758, -0.1365082746823964], { + 9, 13} -> + Complex[-0.15558917647409173`, -0.09979098803090053], {9, + 14} -> Complex[0.1899783589314758, -0.1365082746823964], { + 9, 15} -> + Complex[-8.673617379884035*^-17, \ +-8.326672684688674*^-17], {9, 16} -> Complex[0.3800250972969439, 0.], {9, 17} -> + Complex[0.08485061223426596, -0.11980477813807357`], {9, + 18} -> Complex[ + 0.17479296155780188`, -0.14732776197358494`], {9, 19} -> + Complex[0.13476314496413244`, -0.0767423516894553], {9, + 20} -> Complex[ + 0.02081012384838236, -0.14234469229568147`], {9, 21} -> + Complex[-0.04368290380765367, -0.027436459786564067`], {9, + 22} -> Complex[-0.15157725194981503`, + 0.17844414940708186`], {9, 23} -> + Complex[0.01747597846754094, 0.10203270358947614`], {9, + 24} -> Complex[ + 0.34087862056969404`, 0.1100388283919505], {10, 11} -> + Complex[0.13800495411149877`, -0.3680867713233184], {10, + 12} -> Complex[-0.1905043599805688, \ +-0.13762822610920328`], {10, 13} -> + Complex[0.13800495411149877`, -0.3680867713233184], {10, + 14} -> Complex[-0.1905043599805688, \ +-0.13762822610920328`], {10, 15} -> Complex[0.619974902703056, 0.], {10, 16} -> + Complex[-8.673617379884035*^-17, + 8.326672684688674*^-17], {10, 17} -> + Complex[-0.17479296155780177`, -0.14732776197358502`], { + 10, 18} -> + Complex[-0.08485061223426603, -0.11980477813807351`], {10, + 19} -> Complex[ + 0.12289396168675226`, 0.38209796458606343`], {10, 20} -> + Complex[-0.09460977915945387, -0.11933166053315665`], {10, + 21} -> Complex[ + 0.08210404404737587, 0.05307307167276523], {10, 22} -> + Complex[-0.2346066684020043, -0.5335785389713812], {10, + 23} -> Complex[-0.2917595352077467, 0.3784536838475334], { + 10, 24} -> + Complex[0.33333532926317844`, 0.0052698415932301845`], { + 11, 12} -> Complex[0.5275300911030236, 0.], {11, 13} -> + Complex[1.8041124150158796`*^-16, + 1.457167719820518*^-16], {11, 14} -> + Complex[0.5275300911030236, 0.], {11, 15} -> + Complex[-0.1905043599805688, 0.13762822610920328`], {11, + 16} -> Complex[0.1899783589314758, 0.1365082746823964], { + 11, 17} -> + Complex[-0.04249050665712762, 0.090268967382963], {11, + 18} -> Complex[ + 0.11065480592987563`, 0.022498803837048064`], {11, 19} -> + Complex[-0.19299966150123138`, -0.3282026155242766], {11, + 20} -> Complex[ + 0.017368709836144688`, -0.1497980533645717], {11, 21} -> + Complex[-0.008146862343863803, 0.10273781027166022`], {11, + 22} -> Complex[-0.06250476454855013, + 0.34969666557353624`], {11, 23} -> + Complex[0.11666092373220538`, -0.218936695047531], {11, + 24} -> Complex[ + 0.161016937967343, 0.024863370726720582`], {12, 13} -> + Complex[0.4724699088969763, 0.], {12, 14} -> + Complex[ + 1.8041124150158796`*^-16, -1.457167719820518*^-16], {12, + 15} -> Complex[ + 0.13800495411149877`, 0.3680867713233184], {12, 16} -> + Complex[-0.15558917647409173`, 0.09979098803090053], {12, 17} -> Complex[ - 0.14880448551485936`, 0.18343023977218464`], {10, 18} -> - Complex[0.06911881436797135, -0.04269376157510484], {10, - 19} -> Complex[ - 0.0806509155236052, -0.06238441832735686], {10, 20} -> - Complex[-0.08181885113853848, 0.0223150746441549], {10, + 0.09710709374348331, -0.20541302638200343`], {12, 18} -> + Complex[-0.13183382282185513`, 0.15967490110950291`], {12, + 19} -> Complex[-0.3239368353423375, + 0.27853835234466173`], {12, 20} -> + Complex[0.13688251203715338`, -0.1036078618111689], {12, 21} -> Complex[ - 0.3146513247958228, -0.026797378998685423`], {10, 22} -> - Complex[0.011359633728347016`, -0.08520800008182422], {10, - 23} -> Complex[ - 0.17837001887568327`, 0.09187489939370964], {10, 24} -> - Complex[-0.07164664525064873, 0.2907134789193454], {11, - 12} -> Complex[0.5794387931418044, 0.], {11, 13} -> - Complex[-0.4009452410407828, -0.11284019268466222`], {11, - 14} -> Complex[0.5794387931418044, 0.], {11, 15} -> - Complex[0.3689066714841379, 0.01963818646965646], {11, - 16} -> Complex[-0.1634450723422762, - 0.0011076658174966336`], {11, 17} -> - Complex[0.4351121808870284, 0.1282880779202612], {11, 18} -> - Complex[-0.06875971685231415, -0.006550328119388341], {11, - 19} -> Complex[-0.18425433538775948`, - 0.03665899040288985], {11, 20} -> - Complex[-0.04214475287108173, -0.2125642839376449], {11, - 21} -> Complex[ - 0.5691291778553188, 0.009546455336403484], {11, 22} -> - Complex[-0.11604982216622163`, -0.19926182212673743`], { - 11, 23} -> - Complex[-0.20476687273928368`, -0.20940927658482447`], { - 11, 24} -> - Complex[-0.04633762187626639, 0.37156097014999256`], {12, - 13} -> Complex[0.3744935743083003, 0.], {12, 14} -> - Complex[-0.4009452410407828, 0.11284019268466222`], {12, - 15} -> Complex[-0.364332505630415, - 0.055629112414147225`], {12, 16} -> - Complex[0.24390216997712094`, -0.12967162455970677`], {12, - 17} -> - Complex[-0.2300764357164763, -0.11137567802408056`], {12, - 18} -> Complex[-0.010533198723196371`, - 0.006296053518233197], {12, 19} -> - Complex[-0.029186234486845222`, 0.022372160811117286`], { - 12, 20} -> - Complex[0.14142597329959572`, 0.061350391806736695`], {12, - 21} -> Complex[-0.33684082460892883`, - 0.09005315780013691], {12, 22} -> - Complex[0.09059534447844704, 0.10345647613127328`], {12, - 23} -> Complex[-0.008851768247553363, \ --0.029068202106085572`], {12, 24} -> - Complex[-0.07795077274116374, -0.2904671042221048], {13, - 14} -> Complex[0.5794387931418044, 0.], {13, 15} -> - Complex[0.3689066714841379, 0.01963818646965646], {13, - 16} -> Complex[-0.1634450723422762, - 0.0011076658174966336`], {13, 17} -> - Complex[0.4351121808870284, 0.1282880779202612], {13, 18} -> - Complex[-0.06875971685231415, -0.006550328119388341], {13, - 19} -> Complex[-0.18425433538775948`, - 0.03665899040288985], {13, 20} -> - Complex[-0.04214475287108173, -0.2125642839376449], {13, - 21} -> Complex[ - 0.5691291778553188, 0.009546455336403484], {13, 22} -> - Complex[-0.11604982216622163`, -0.19926182212673743`], { - 13, 23} -> - Complex[-0.20476687273928368`, -0.20940927658482447`], { - 13, 24} -> - Complex[-0.04633762187626639, 0.37156097014999256`], {14, - 15} -> Complex[-0.364332505630415, - 0.055629112414147225`], {14, 16} -> - Complex[0.24390216997712094`, -0.12967162455970677`], {14, - 17} -> - Complex[-0.2300764357164763, -0.11137567802408056`], {14, - 18} -> Complex[-0.010533198723196371`, - 0.006296053518233197], {14, 19} -> - Complex[-0.029186234486845222`, 0.022372160811117286`], { - 14, 20} -> - Complex[0.14142597329959572`, 0.061350391806736695`], {14, - 21} -> Complex[-0.33684082460892883`, - 0.09005315780013691], {14, 22} -> - Complex[0.09059534447844704, 0.10345647613127328`], {14, - 23} -> Complex[-0.008851768247553363, \ --0.029068202106085572`], {14, 24} -> - Complex[-0.07795077274116374, -0.2904671042221048], {15, - 16} -> Complex[0.41416215192864014`, 0.], {15, 17} -> - Complex[0.1868664022204954, -0.06385028540854319], {15, - 18} -> Complex[-0.11463533571593523`, \ --0.055210078698593804`], {15, 19} -> - Complex[-0.2934854494126851, -0.04085502210736608], {15, + 0.01476498538421066, -0.0004105289599095289], {12, 22} -> + Complex[0.25833102475550235`, -0.37839808713437206`], {12, + 23} -> Complex[-0.0868392972934217, + 0.014513008624828502`], {12, 24} -> + Complex[-0.16585075463443222`, 0.4085141858532183], {13, + 14} -> Complex[0.5275300911030236, 0.], {13, 15} -> + Complex[-0.1905043599805688, 0.13762822610920328`], {13, + 16} -> Complex[0.1899783589314758, 0.1365082746823964], { + 13, 17} -> + Complex[-0.04249050665712762, 0.090268967382963], {13, + 18} -> Complex[ + 0.11065480592987563`, 0.022498803837048064`], {13, 19} -> + Complex[-0.19299966150123138`, -0.3282026155242766], {13, 20} -> Complex[ - 0.22168267144312054`, -0.03299920735716682], {15, 21} -> - Complex[-0.06237372995183321, 0.06727056435677253], {15, - 22} -> Complex[-0.030346695869449408`, \ --0.030149558835644692`], {15, 23} -> - Complex[-0.09727281618814543, -0.4251726947214757], {15, - 24} -> Complex[0.02023138254715448, -0.11137525641609], { - 16, 17} -> - Complex[0.14880448551485936`, 0.18343023977218464`], {16, + 0.017368709836144688`, -0.1497980533645717], {13, 21} -> + Complex[-0.008146862343863803, 0.10273781027166022`], {13, + 22} -> Complex[-0.06250476454855013, + 0.34969666557353624`], {13, 23} -> + Complex[0.11666092373220538`, -0.218936695047531], {13, + 24} -> Complex[ + 0.161016937967343, 0.024863370726720582`], {14, 15} -> + Complex[0.13800495411149877`, 0.3680867713233184], {14, + 16} -> Complex[-0.15558917647409173`, + 0.09979098803090053], {14, 17} -> + Complex[0.09710709374348331, -0.20541302638200343`], {14, + 18} -> Complex[-0.13183382282185513`, + 0.15967490110950291`], {14, 19} -> + Complex[-0.3239368353423375, 0.27853835234466173`], {14, + 20} -> Complex[ + 0.13688251203715338`, -0.1036078618111689], {14, 21} -> + Complex[0.01476498538421066, -0.0004105289599095289], {14, + 22} -> Complex[ + 0.25833102475550235`, -0.37839808713437206`], {14, 23} -> + Complex[-0.0868392972934217, 0.014513008624828502`], {14, + 24} -> Complex[-0.16585075463443222`, + 0.4085141858532183], {15, 16} -> + Complex[0.3800250972969439, 0.], {15, 17} -> + Complex[0.08485061223426596, -0.11980477813807357`], {15, 18} -> Complex[ - 0.06911881436797135, -0.04269376157510484], {16, 19} -> - Complex[0.0806509155236052, -0.06238441832735686], {16, - 20} -> Complex[-0.08181885113853848, - 0.0223150746441549], {16, 21} -> - Complex[0.3146513247958228, -0.026797378998685423`], {16, - 22} -> Complex[ - 0.011359633728347016`, -0.08520800008182422], {16, 23} -> - Complex[0.17837001887568327`, 0.09187489939370964], {16, - 24} -> Complex[-0.07164664525064873, - 0.2907134789193454], {17, 18} -> - Complex[0.09630860197729411, 0.], {17, 19} -> - Complex[0.12702736104247997`, -0.01646149694625006], {17, - 20} -> Complex[-0.03764992141799037, - 0.13605099778794436`], {17, 21} -> - Complex[-0.07115780937991523, -0.011205895451017422`], { + 0.17479296155780188`, -0.14732776197358494`], {15, 19} -> + Complex[0.13476314496413244`, -0.0767423516894553], {15, + 20} -> Complex[ + 0.02081012384838236, -0.14234469229568147`], {15, 21} -> + Complex[-0.04368290380765367, -0.027436459786564067`], { + 15, 22} -> + Complex[-0.15157725194981503`, 0.17844414940708186`], {15, + 23} -> Complex[ + 0.01747597846754094, 0.10203270358947614`], {15, 24} -> + Complex[0.34087862056969404`, 0.1100388283919505], {16, + 17} -> Complex[-0.17479296155780177`, \ +-0.14732776197358502`], {16, 18} -> + Complex[-0.08485061223426603, -0.11980477813807351`], {16, + 19} -> Complex[ + 0.12289396168675226`, 0.38209796458606343`], {16, 20} -> + Complex[-0.09460977915945387, -0.11933166053315665`], {16, + 21} -> Complex[ + 0.08210404404737587, 0.05307307167276523], {16, 22} -> + Complex[-0.2346066684020043, -0.5335785389713812], {16, + 23} -> Complex[-0.2917595352077467, 0.3784536838475334], { + 16, 24} -> + Complex[0.33333532926317844`, 0.0052698415932301845`], { + 17, 18} -> Complex[0.5226137655938699, 0.], {17, 19} -> + Complex[0.2578656808252187, 0.10876030801336257`], {17, + 20} -> Complex[-0.007671097858546656, + 0.0057340892009139326`], {17, 21} -> + Complex[-0.13167457410588743`, -0.003054508743256395], { 17, 22} -> - Complex[0.07416506719537758, 0.08051189278847616], {17, - 23} -> Complex[0.16774613988004144`, 0.225655277072902], { + Complex[0.037739555243020126`, -0.012308050579545225`], { + 17, 23} -> + Complex[-0.23097227051538433`, -0.42320695740719316`], { 17, 24} -> - Complex[-0.09714752480970831, -0.05391368675976281], {18, - 19} -> Complex[-0.4024303768625533, \ --0.06319936655428628], {18, 20} -> - Complex[-0.02707098756043888, -0.08415309369657159], {18, - 21} -> Complex[0.6164020052722983, 0.03301684548082712], { - 18, 22} -> - Complex[-0.2385656438458264, -0.06903700176549488], {18, - 23} -> Complex[-0.24737746704016716`, \ --0.5022802985841684], {18, 24} -> - Complex[0.199408104684111, 0.11720650121625686`], {19, - 20} -> Complex[0.460087813322552, 0.], {19, 21} -> - Complex[-0.020878723063805726`, 0.1044611112500656], {19, - 22} -> Complex[ - 0.24014852879671952`, -0.15329539227786965`], {19, 23} -> - Complex[0.11519091687891489`, -0.4012928958026726], {19, - 24} -> Complex[-0.45588071484204273`, - 0.014752170920114424`], {20, 21} -> - Complex[-0.32198385415100245`, -0.056214448752933965`], { - 20, 22} -> - Complex[0.02087872306380592, 0.10446111125006569`], {20, - 23} -> Complex[ - 0.34202714802636813`, 0.5203828231210746], {20, 24} -> - Complex[0.27130379014783124`, 0.12044715596755934`], {21, - 22} -> Complex[0.2677162255621613, 0.], {21, 23} -> - Complex[0.27244326195988916`, -0.011111878577649018`], { - 21, 24} -> - Complex[-0.3506829757791454, -0.10726001244941838`], {22, - 23} -> Complex[-0.29340373279675197`, \ --0.27809938962030384`], {22, 24} -> - Complex[-0.07968743240634073, 0.17702649633258472`]}, + Complex[0.29687889692943875`, 0.2375326567068352], {18, + 19} -> Complex[-0.17834040611966948`, + 0.12609948481270494`], {18, 20} -> + Complex[0.22516518597054483`, 0.1013577129442031], {18, + 21} -> Complex[ + 0.053161902759652525`, -0.14451247653181823`], {18, 22} -> + Complex[0.2616855917631509, -0.009667937264448562], {18, + 23} -> Complex[-0.16418162796222252`, + 0.13133580222113328`], {18, 24} -> + Complex[-0.42262163352679005`, 0.31456841745062747`], {19, + 20} -> Complex[0.21640487397840627`, 0.], {19, 21} -> + Complex[-0.029720518023860276`, -0.053544088868121745`], { + 19, 22} -> Complex[ + 0.030636763457055183`, -0.11355874224143539`], {19, 23} -> + Complex[-0.06797212261961828, 0.14596193524618625`], {19, + 24} -> Complex[-0.1599463611986579, 0.3186964622408383], { + 20, 21} -> + Complex[-0.15211930857597808`, -0.0462428201376506], {20, + 22} -> Complex[-0.5748910514001031, \ +-0.14633045383134305`], {20, 23} -> + Complex[0.029019428968993538`, 0.21528502386671927`], {20, + 24} -> Complex[ + 0.46472596940822686`, 0.06199143840307167], {21, 22} -> + Complex[0.8762699833354937, 0.], {21, 23} -> + 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Complex[-0.2385656438458264, -0.06903700176549488], {18, 23} -> - Complex[-0.24737746704016716`, -0.5022802985841684], {18, 24} -> - Complex[0.199408104684111, 0.11720650121625686`], {19, 20} -> - Complex[0.460087813322552, 0.], {19, 21} -> - Complex[-0.020878723063805726`, 0.1044611112500656], {19, 22} -> - Complex[0.24014852879671952`, -0.15329539227786965`], {19, 23} -> - Complex[0.11519091687891489`, -0.4012928958026726], {19, 24} -> - Complex[-0.45588071484204273`, 0.014752170920114424`], {20, 21} -> - Complex[-0.32198385415100245`, -0.056214448752933965`], {20, 22} -> - Complex[0.02087872306380592, 0.10446111125006569`], {20, 23} -> - Complex[0.34202714802636813`, 0.5203828231210746], {20, 24} -> - Complex[0.27130379014783124`, 0.12044715596755934`], {21, 22} -> - Complex[0.2677162255621613, 0.], {21, 23} -> - Complex[0.27244326195988916`, -0.011111878577649018`], {21, 24} -> - Complex[-0.3506829757791454, -0.10726001244941838`], {22, 23} -> - Complex[-0.29340373279675197`, -0.27809938962030384`], {22, 24} -> - Complex[-0.07968743240634073, 0.17702649633258472`]}, + Complex[-0.06947320790243908, -0.12537107773431694`], {2, 4} -> + Complex[-0.2782895722096578, 0.5061420791848172], {2, 5} -> + Complex[0.029019428968993538`, -0.21528502386671927`], {2, 6} -> + Complex[-0.06797212261961828, -0.14596193524618625`], {2, 7} -> + Complex[-0.16418162796222252`, -0.13133580222113328`], {2, 8} -> + Complex[-0.23097227051538433`, 0.42320695740719316`], {2, 9} -> + Complex[-0.2917595352077467, -0.3784536838475334], {2, 10} -> + Complex[0.01747597846754094, -0.10203270358947614`], {2, 11} -> + Complex[-0.0868392972934217, -0.014513008624828502`], {2, 12} -> + Complex[0.11666092373220538`, 0.218936695047531], {2, 13} -> + Complex[-0.0868392972934217, -0.014513008624828502`], {2, 14} -> + Complex[0.11666092373220538`, 0.218936695047531], {2, 15} -> + Complex[-0.2917595352077467, -0.3784536838475334], {2, 16} -> + Complex[0.01747597846754094, -0.10203270358947614`], {2, 17} -> + Complex[-0.16418162796222252`, -0.13133580222113328`], {2, 18} -> + Complex[-0.23097227051538433`, 0.42320695740719316`], {2, 19} -> + Complex[0.029019428968993538`, -0.21528502386671927`], {2, 20} -> + Complex[-0.06797212261961828, -0.14596193524618625`], {2, 21} -> + Complex[-0.06947320790243908, -0.12537107773431694`], {2, 22} -> + Complex[-0.2782895722096578, 0.5061420791848172], {2, 23} -> + 1, {3, 4} -> Complex[0.8762699833354937, 0.], {3, 5} -> + Complex[-0.5748910514001031, 0.14633045383134305`], {3, 6} -> + Complex[0.030636763457055183`, 0.11355874224143539`], {3, 7} -> + Complex[0.2616855917631509, 0.009667937264448562], {3, 8} -> + Complex[0.037739555243020126`, 0.012308050579545225`], {3, 9} -> + Complex[-0.2346066684020043, 0.5335785389713812], {3, 10} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {3, 11} -> + Complex[0.25833102475550235`, 0.37839808713437206`], {3, 12} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {3, 13} -> + Complex[0.25833102475550235`, 0.37839808713437206`], {3, 14} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {3, 15} -> + Complex[-0.2346066684020043, 0.5335785389713812], {3, 16} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {3, 17} -> + Complex[0.2616855917631509, 0.009667937264448562], {3, 18} -> + Complex[0.037739555243020126`, 0.012308050579545225`], {3, 19} -> + Complex[-0.5748910514001031, 0.14633045383134305`], {3, 20} -> + Complex[0.030636763457055183`, 0.11355874224143539`], {3, 21} -> + Complex[-1.1102230246251565`*^-16, 0.], {3, 22} -> + Complex[0.8762699833354937, 0.], {3, 23} -> + Complex[-0.2782895722096578, -0.5061420791848172], {3, 24} -> + Complex[-0.41976082253431385`, 0.08596477743313696], {4, 5} -> + Complex[-0.15211930857597808`, 0.0462428201376506], {4, 6} -> + Complex[-0.029720518023860276`, 0.053544088868121745`], {4, 7} -> + Complex[0.053161902759652525`, 0.14451247653181823`], {4, 8} -> + Complex[-0.13167457410588743`, 0.003054508743256395], {4, 9} -> + Complex[0.08210404404737587, -0.05307307167276523], {4, 10} -> + Complex[-0.04368290380765367, 0.027436459786564067`], {4, 11} -> + Complex[0.01476498538421066, 0.0004105289599095289], {4, 12} -> + Complex[-0.008146862343863803, -0.10273781027166022`], {4, 13} -> + Complex[0.01476498538421066, 0.0004105289599095289], {4, 14} -> + Complex[-0.008146862343863803, -0.10273781027166022`], {4, 15} -> + Complex[0.08210404404737587, -0.05307307167276523], {4, 16} -> + Complex[-0.04368290380765367, 0.027436459786564067`], {4, 17} -> + Complex[0.053161902759652525`, 0.14451247653181823`], {4, 18} -> + Complex[-0.13167457410588743`, 0.003054508743256395], {4, 19} -> + Complex[-0.15211930857597808`, 0.0462428201376506], {4, 20} -> + Complex[-0.029720518023860276`, 0.053544088868121745`], {4, 21} -> + Complex[0.12373001666450631`, 0.], {4, 22} -> + Complex[-1.1102230246251565`*^-16, 0.], {4, 23} -> + Complex[-0.06947320790243908, 0.12537107773431694`], {4, 24} -> + Complex[-0.1766768302665536, -0.1352957631030186], {5, 6} -> + Complex[0.21640487397840627`, 0.], {5, 7} -> + Complex[0.22516518597054483`, -0.1013577129442031], {5, 8} -> + Complex[-0.007671097858546656, -0.0057340892009139326`], {5, 9} -> + Complex[-0.09460977915945387, 0.11933166053315665`], {5, 10} -> + Complex[0.02081012384838236, 0.14234469229568147`], {5, 11} -> + Complex[0.13688251203715338`, 0.1036078618111689], {5, 12} -> + Complex[0.017368709836144688`, 0.1497980533645717], {5, 13} -> + Complex[0.13688251203715338`, 0.1036078618111689], {5, 14} -> + Complex[0.017368709836144688`, 0.1497980533645717], {5, 15} -> + Complex[-0.09460977915945387, 0.11933166053315665`], {5, 16} -> + Complex[0.02081012384838236, 0.14234469229568147`], {5, 17} -> + Complex[0.22516518597054483`, -0.1013577129442031], {5, 18} -> + Complex[-0.007671097858546656, -0.0057340892009139326`], {5, 19} -> + Complex[8.326672684688674*^-17, 1.3877787807814457`*^-17], {5, 20} -> + Complex[0.21640487397840627`, 0.], {5, 21} -> + Complex[-0.029720518023860276`, -0.053544088868121745`], {5, 22} -> + Complex[0.030636763457055183`, -0.11355874224143539`], {5, 23} -> + Complex[-0.06797212261961828, 0.14596193524618625`], {5, 24} -> + Complex[-0.1599463611986579, 0.3186964622408383], {6, 7} -> + Complex[-0.17834040611966948`, -0.12609948481270494`], {6, 8} -> + Complex[0.2578656808252187, -0.10876030801336257`], {6, 9} -> + Complex[0.12289396168675226`, -0.38209796458606343`], {6, 10} -> + Complex[0.13476314496413244`, 0.0767423516894553], {6, 11} -> + Complex[-0.3239368353423375, -0.27853835234466173`], {6, 12} -> + Complex[-0.19299966150123138`, 0.3282026155242766], {6, 13} -> + Complex[-0.3239368353423375, -0.27853835234466173`], {6, 14} -> + Complex[-0.19299966150123138`, 0.3282026155242766], {6, 15} -> + Complex[0.12289396168675226`, -0.38209796458606343`], {6, 16} -> + Complex[0.13476314496413244`, 0.0767423516894553], {6, 17} -> + Complex[-0.17834040611966948`, -0.12609948481270494`], {6, 18} -> + Complex[0.2578656808252187, -0.10876030801336257`], {6, 19} -> + Complex[0.7835951260215941, 0.], {6, 20} -> + Complex[8.326672684688674*^-17, -1.3877787807814457`*^-17], {6, + 21} -> Complex[-0.15211930857597808`, -0.0462428201376506], {6, + 22} -> Complex[-0.5748910514001031, -0.14633045383134305`], {6, + 23} -> Complex[0.029019428968993538`, 0.21528502386671927`], {6, + 24} -> Complex[0.46472596940822686`, 0.06199143840307167], {7, 8} -> + Complex[0.5226137655938699, 0.], {7, 9} -> + Complex[-0.08485061223426603, 0.11980477813807351`], {7, 10} -> + Complex[0.17479296155780188`, 0.14732776197358494`], {7, 11} -> + Complex[-0.13183382282185513`, -0.15967490110950291`], {7, 12} -> + Complex[0.11065480592987563`, -0.022498803837048064`], {7, 13} -> + Complex[-0.13183382282185513`, -0.15967490110950291`], {7, 14} -> + Complex[0.11065480592987563`, -0.022498803837048064`], {7, 15} -> + Complex[-0.08485061223426603, 0.11980477813807351`], {7, 16} -> + Complex[0.17479296155780188`, 0.14732776197358494`], {7, 17} -> + Complex[-1.7867651802561116`*^-16, 1.1449174941446927`*^-16], {7, + 18} -> Complex[0.5226137655938699, 0.], {7, 19} -> + Complex[0.2578656808252187, 0.10876030801336257`], {7, 20} -> + Complex[-0.007671097858546656, 0.0057340892009139326`], {7, 21} -> + Complex[-0.13167457410588743`, -0.003054508743256395], {7, 22} -> + Complex[0.037739555243020126`, -0.012308050579545225`], {7, 23} -> + Complex[-0.23097227051538433`, -0.42320695740719316`], {7, 24} -> + Complex[0.29687889692943875`, 0.2375326567068352], {8, 9} -> + Complex[-0.17479296155780177`, 0.14732776197358502`], {8, 10} -> + Complex[0.08485061223426596, 0.11980477813807357`], {8, 11} -> + Complex[0.09710709374348331, 0.20541302638200343`], {8, 12} -> + Complex[-0.04249050665712762, -0.090268967382963], {8, 13} -> + Complex[0.09710709374348331, 0.20541302638200343`], {8, 14} -> + Complex[-0.04249050665712762, -0.090268967382963], {8, 15} -> + Complex[-0.17479296155780177`, 0.14732776197358502`], {8, 16} -> + Complex[0.08485061223426596, 0.11980477813807357`], {8, 17} -> + Complex[0.47738623440613004`, 0.], {8, 18} -> + Complex[-1.7867651802561116`*^-16, -1.1449174941446927`*^-16], {8, + 19} -> Complex[-0.17834040611966948`, 0.12609948481270494`], {8, + 20} -> Complex[0.22516518597054483`, 0.1013577129442031], {8, 21} -> + Complex[0.053161902759652525`, -0.14451247653181823`], {8, 22} -> + Complex[0.2616855917631509, -0.009667937264448562], {8, 23} -> + Complex[-0.16418162796222252`, 0.13133580222113328`], {8, 24} -> + Complex[-0.42262163352679005`, 0.31456841745062747`], {9, 10} -> + Complex[0.3800250972969439, 0.], {9, 11} -> + Complex[-0.15558917647409173`, -0.09979098803090053], {9, 12} -> + Complex[0.1899783589314758, -0.1365082746823964], {9, 13} -> + Complex[-0.15558917647409173`, -0.09979098803090053], {9, 14} -> + Complex[0.1899783589314758, -0.1365082746823964], {9, 15} -> + Complex[-8.673617379884035*^-17, -8.326672684688674*^-17], {9, 16} -> + Complex[0.3800250972969439, 0.], {9, 17} -> + Complex[0.08485061223426596, -0.11980477813807357`], {9, 18} -> + Complex[0.17479296155780188`, -0.14732776197358494`], {9, 19} -> + Complex[0.13476314496413244`, -0.0767423516894553], {9, 20} -> + Complex[0.02081012384838236, -0.14234469229568147`], {9, 21} -> + Complex[-0.04368290380765367, -0.027436459786564067`], {9, 22} -> + Complex[-0.15157725194981503`, 0.17844414940708186`], {9, 23} -> + Complex[0.01747597846754094, 0.10203270358947614`], {9, 24} -> + Complex[0.34087862056969404`, 0.1100388283919505], {10, 11} -> + Complex[0.13800495411149877`, -0.3680867713233184], {10, 12} -> + Complex[-0.1905043599805688, -0.13762822610920328`], {10, 13} -> + Complex[0.13800495411149877`, -0.3680867713233184], {10, 14} -> + Complex[-0.1905043599805688, -0.13762822610920328`], {10, 15} -> + Complex[0.619974902703056, 0.], {10, 16} -> + Complex[-8.673617379884035*^-17, 8.326672684688674*^-17], {10, 17} -> + Complex[-0.17479296155780177`, -0.14732776197358502`], {10, 18} -> + Complex[-0.08485061223426603, -0.11980477813807351`], {10, 19} -> + Complex[0.12289396168675226`, 0.38209796458606343`], {10, 20} -> + Complex[-0.09460977915945387, -0.11933166053315665`], {10, 21} -> + Complex[0.08210404404737587, 0.05307307167276523], {10, 22} -> + Complex[-0.2346066684020043, -0.5335785389713812], {10, 23} -> + Complex[-0.2917595352077467, 0.3784536838475334], {10, 24} -> + Complex[0.33333532926317844`, 0.0052698415932301845`], {11, 12} -> + Complex[0.5275300911030236, 0.], {11, 13} -> + Complex[1.8041124150158796`*^-16, 1.457167719820518*^-16], {11, + 14} -> Complex[0.5275300911030236, 0.], {11, 15} -> + Complex[-0.1905043599805688, 0.13762822610920328`], {11, 16} -> + Complex[0.1899783589314758, 0.1365082746823964], {11, 17} -> + Complex[-0.04249050665712762, 0.090268967382963], {11, 18} -> + Complex[0.11065480592987563`, 0.022498803837048064`], {11, 19} -> + Complex[-0.19299966150123138`, -0.3282026155242766], {11, 20} -> + Complex[0.017368709836144688`, -0.1497980533645717], {11, 21} -> + Complex[-0.008146862343863803, 0.10273781027166022`], {11, 22} -> + Complex[-0.06250476454855013, 0.34969666557353624`], {11, 23} -> + Complex[0.11666092373220538`, -0.218936695047531], {11, 24} -> + Complex[0.161016937967343, 0.024863370726720582`], {12, 13} -> + Complex[0.4724699088969763, 0.], {12, 14} -> + Complex[1.8041124150158796`*^-16, -1.457167719820518*^-16], {12, + 15} -> Complex[0.13800495411149877`, 0.3680867713233184], {12, + 16} -> Complex[-0.15558917647409173`, 0.09979098803090053], {12, + 17} -> Complex[0.09710709374348331, -0.20541302638200343`], {12, + 18} -> Complex[-0.13183382282185513`, 0.15967490110950291`], {12, + 19} -> Complex[-0.3239368353423375, 0.27853835234466173`], {12, + 20} -> Complex[0.13688251203715338`, -0.1036078618111689], {12, + 21} -> Complex[0.01476498538421066, -0.0004105289599095289], {12, + 22} -> Complex[0.25833102475550235`, 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{3, 13} -> + Complex[0.01476498538421066, 0.0004105289599095289], {3, + 14} -> Complex[-0.008146862343863803, \ +-0.10273781027166022`], {3, 15} -> + Complex[-0.04368290380765367, 0.027436459786564067`], {3, + 16} -> Complex[ + 0.08210404404737587, -0.05307307167276523], {3, 17} -> + Complex[-0.13167457410588743`, 0.003054508743256395], {3, + 18} -> Complex[ + 0.053161902759652525`, 0.14451247653181823`], {3, 19} -> + Complex[-0.15211930857597808`, 0.0462428201376506], {3, + 20} -> Complex[-0.029720518023860276`, + 0.053544088868121745`], {3, 21} -> + Complex[-1.1102230246251565`*^-16, 0.], {3, 22} -> + Complex[0.12373001666450631`, 0.], {3, 23} -> + Complex[-0.06947320790243908, 0.12537107773431694`], {3, + 24} -> Complex[-0.1766768302665536, -0.1352957631030186], \ +{4, 5} -> Complex[-0.5748910514001031, 0.14633045383134305`], {4, 6} -> + Complex[0.030636763457055183`, 0.11355874224143539`], {4, + 7} -> Complex[ + 0.037739555243020126`, 0.012308050579545225`], {4, 8} -> + Complex[0.2616855917631509, 0.009667937264448562], {4, 9} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {4, + 10} -> Complex[-0.2346066684020043, + 0.5335785389713812], {4, 11} -> + Complex[0.25833102475550235`, 0.37839808713437206`], {4, + 12} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {4, + 13} -> Complex[ + 0.25833102475550235`, 0.37839808713437206`], {4, 14} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {4, + 15} -> Complex[-0.15157725194981503`, \ +-0.17844414940708186`], {4, 16} -> + Complex[-0.2346066684020043, 0.5335785389713812], {4, 17} -> + Complex[0.037739555243020126`, 0.012308050579545225`], {4, + 18} -> Complex[ + 0.2616855917631509, 0.009667937264448562], {4, 19} -> + Complex[-0.5748910514001031, 0.14633045383134305`], {4, + 20} -> Complex[ + 0.030636763457055183`, 0.11355874224143539`], {4, 21} -> + Complex[0.8762699833354937, 0.], {4, 22} -> + Complex[-1.1102230246251565`*^-16, 0.], {4, 23} -> + Complex[-0.2782895722096578, 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1.3877787807814457`*^-17], {5, 20} -> + Complex[0.21640487397840627`, 0.], {5, 21} -> + Complex[0.030636763457055183`, -0.11355874224143539`], {5, + 22} -> + Complex[-0.029720518023860276`, -0.053544088868121745`], { + 5, 23} -> Complex[-0.06797212261961828, + 0.14596193524618625`], {5, 24} -> + Complex[-0.1599463611986579, 0.3186964622408383], {6, 7} -> + Complex[0.2578656808252187, -0.10876030801336257`], {6, + 8} -> Complex[-0.17834040611966948`, \ +-0.12609948481270494`], {6, 9} -> + Complex[0.13476314496413244`, 0.0767423516894553], {6, + 10} -> Complex[ + 0.12289396168675226`, -0.38209796458606343`], {6, 11} -> + Complex[-0.3239368353423375, -0.27853835234466173`], {6, + 12} -> Complex[-0.19299966150123138`, + 0.3282026155242766], {6, 13} -> + Complex[-0.3239368353423375, -0.27853835234466173`], {6, + 14} -> Complex[-0.19299966150123138`, + 0.3282026155242766], {6, 15} -> + Complex[0.13476314496413244`, 0.0767423516894553], {6, + 16} -> Complex[ + 0.12289396168675226`, -0.38209796458606343`], {6, 17} -> + Complex[0.2578656808252187, -0.10876030801336257`], {6, + 18} -> Complex[-0.17834040611966948`, \ +-0.12609948481270494`], {6, 19} -> Complex[0.7835951260215941, 0.], {6, 20} -> + Complex[ + 8.326672684688674*^-17, -1.3877787807814457`*^-17], {6, + 21} -> Complex[-0.5748910514001031, \ +-0.14633045383134305`], {6, 22} -> + Complex[-0.15211930857597808`, -0.0462428201376506], {6, + 23} -> Complex[ + 0.029019428968993538`, 0.21528502386671927`], {6, 24} -> + Complex[0.46472596940822686`, 0.06199143840307167], {7, + 8} -> Complex[0.47738623440613004`, 0.], {7, 9} -> + Complex[0.08485061223426596, 0.11980477813807357`], {7, + 10} -> Complex[-0.17479296155780177`, + 0.14732776197358502`], {7, 11} -> + Complex[0.09710709374348331, 0.20541302638200343`], {7, + 12} -> Complex[-0.04249050665712762, -0.090268967382963], \ +{7, 13} -> Complex[0.09710709374348331, 0.20541302638200343`], {7, 14} -> + Complex[-0.04249050665712762, -0.090268967382963], {7, + 15} -> Complex[ + 0.08485061223426596, 0.11980477813807357`], {7, 16} -> + Complex[-0.17479296155780177`, 0.14732776197358502`], {7, + 17} -> Complex[-1.7867651802561116`*^-16, \ +-1.1449174941446927`*^-16], {7, 18} -> + Complex[0.47738623440613004`, 0.], {7, 19} -> + Complex[-0.17834040611966948`, 0.12609948481270494`], {7, + 20} -> Complex[ + 0.22516518597054483`, 0.1013577129442031], {7, 21} -> + Complex[0.2616855917631509, -0.009667937264448562], {7, + 22} -> Complex[ + 0.053161902759652525`, -0.14451247653181823`], {7, 23} -> + Complex[-0.16418162796222252`, 0.13133580222113328`], {7, + 24} -> Complex[-0.42262163352679005`, + 0.31456841745062747`], {8, 9} -> + Complex[0.17479296155780188`, 0.14732776197358494`], {8, + 10} -> Complex[-0.08485061223426603, + 0.11980477813807351`], {8, 11} -> + Complex[-0.13183382282185513`, -0.15967490110950291`], {8, + 12} -> Complex[ + 0.11065480592987563`, -0.022498803837048064`], {8, 13} -> + Complex[-0.13183382282185513`, -0.15967490110950291`], 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Complex[0.3800250972969439, 0.], {10, 16} -> + Complex[-8.673617379884035*^-17, \ +-8.326672684688674*^-17], {10, 17} -> + Complex[0.17479296155780188`, -0.14732776197358494`], {10, + 18} -> Complex[ + 0.08485061223426596, -0.11980477813807357`], {10, 19} -> + Complex[0.13476314496413244`, -0.0767423516894553], {10, + 20} -> Complex[ + 0.02081012384838236, -0.14234469229568147`], {10, 21} -> + Complex[-0.15157725194981503`, 0.17844414940708186`], {10, + 22} -> + Complex[-0.04368290380765367, -0.027436459786564067`], { + 10, 23} -> + Complex[0.01747597846754094, 0.10203270358947614`], {10, + 24} -> Complex[ + 0.34087862056969404`, 0.1100388283919505], {11, 12} -> + Complex[0.5275300911030236, 0.], {11, 13} -> + Complex[1.8041124150158796`*^-16, + 1.457167719820518*^-16], {11, 14} -> + Complex[0.5275300911030236, 0.], {11, 15} -> + Complex[0.1899783589314758, 0.1365082746823964], {11, 16} -> + Complex[-0.1905043599805688, 0.13762822610920328`], {11, + 17} -> Complex[ + 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Complex[8.326672684688674*^-17, + 1.3877787807814457`*^-17], {6, 21} -> + Complex[-0.029720518023860276`, -0.053544088868121745`], { + 6, 22} -> Complex[ + 0.030636763457055183`, -0.11355874224143539`], {6, 23} -> + Complex[-0.06797212261961828, 0.14596193524618625`], {6, + 24} -> Complex[-0.1599463611986579, 0.3186964622408383], { + 7, 8} -> Complex[0.47738623440613004`, 0.], {7, 9} -> + Complex[0.08485061223426596, 0.11980477813807357`], {7, + 10} -> Complex[-0.17479296155780177`, + 0.14732776197358502`], {7, 11} -> + Complex[0.09710709374348331, 0.20541302638200343`], {7, + 12} -> Complex[-0.04249050665712762, -0.090268967382963], \ +{7, 13} -> Complex[0.09710709374348331, 0.20541302638200343`], {7, 14} -> + Complex[-0.04249050665712762, -0.090268967382963], {7, + 15} -> Complex[ + 0.08485061223426596, 0.11980477813807357`], {7, 16} -> + Complex[-0.17479296155780177`, 0.14732776197358502`], {7, + 17} -> Complex[-1.7867651802561116`*^-16, \ +-1.1449174941446927`*^-16], {7, 18} -> + 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0.0462428201376506], {3, 21} -> + Complex[-1.1102230246251565`*^-16, 0.], {3, 22} -> + Complex[0.12373001666450631`, 0.], {3, 23} -> + Complex[-0.06947320790243908, 0.12537107773431694`], {3, + 24} -> Complex[-0.1766768302665536, -0.1352957631030186], \ +{4, 5} -> Complex[0.030636763457055183`, 0.11355874224143539`], {4, 6} -> + Complex[-0.5748910514001031, 0.14633045383134305`], {4, + 7} -> Complex[0.2616855917631509, 0.009667937264448562], { + 4, 8} -> Complex[ + 0.037739555243020126`, 0.012308050579545225`], {4, 9} -> + Complex[-0.2346066684020043, 0.5335785389713812], {4, 10} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {4, + 11} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {4, + 12} -> Complex[ + 0.25833102475550235`, 0.37839808713437206`], {4, 13} -> + Complex[-0.06250476454855013, -0.34969666557353624`], {4, + 14} -> Complex[ + 0.25833102475550235`, 0.37839808713437206`], {4, 15} -> + Complex[-0.2346066684020043, 0.5335785389713812], {4, 16} -> + Complex[-0.15157725194981503`, -0.17844414940708186`], {4, + 17} -> Complex[ + 0.2616855917631509, 0.009667937264448562], {4, 18} -> + Complex[0.037739555243020126`, 0.012308050579545225`], {4, + 19} -> Complex[ + 0.030636763457055183`, 0.11355874224143539`], {4, 20} -> + Complex[-0.5748910514001031, 0.14633045383134305`], {4, + 21} -> Complex[0.8762699833354937, 0.], {4, 22} -> + Complex[-1.1102230246251565`*^-16, 0.], {4, 23} -> + Complex[-0.2782895722096578, -0.5061420791848172], {4, + 24} -> Complex[-0.41976082253431385`, + 0.08596477743313696], {5, 6} -> + Complex[0.7835951260215941, 0.], {5, 7} -> + Complex[-0.17834040611966948`, -0.12609948481270494`], {5, + 8} -> Complex[ + 0.2578656808252187, -0.10876030801336257`], {5, 9} -> + Complex[0.12289396168675226`, -0.38209796458606343`], {5, + 10} -> Complex[ + 0.13476314496413244`, 0.0767423516894553], {5, 11} -> + Complex[-0.19299966150123138`, 0.3282026155242766], {5, + 12} -> Complex[-0.3239368353423375, \ +-0.27853835234466173`], {5, 13} -> + Complex[-0.19299966150123138`, 0.3282026155242766], {5, + 14} -> Complex[-0.3239368353423375, \ +-0.27853835234466173`], {5, 15} -> + Complex[0.12289396168675226`, -0.38209796458606343`], {5, + 16} -> Complex[ + 0.13476314496413244`, 0.0767423516894553], {5, 17} -> + Complex[-0.17834040611966948`, -0.12609948481270494`], {5, + 18} -> Complex[ + 0.2578656808252187, -0.10876030801336257`], {5, 19} -> + Complex[ + 8.326672684688674*^-17, -1.3877787807814457`*^-17], {5, + 20} -> Complex[0.7835951260215941, 0.], {5, 21} -> + Complex[-0.5748910514001031, -0.14633045383134305`], {5, + 22} -> Complex[-0.15211930857597808`, \ +-0.0462428201376506], {5, 23} -> + Complex[0.029019428968993538`, 0.21528502386671927`], {5, + 24} -> Complex[ + 0.46472596940822686`, 0.06199143840307167], {6, 7} -> + Complex[0.22516518597054483`, -0.1013577129442031], {6, + 8} -> Complex[-0.007671097858546656, \ +-0.0057340892009139326`], {6, 9} -> + Complex[-0.09460977915945387, 0.11933166053315665`], {6, + 10} -> Complex[ + 0.02081012384838236, 0.14234469229568147`], {6, 11} -> + Complex[0.017368709836144688`, 0.1497980533645717], {6, + 12} -> Complex[ + 0.13688251203715338`, 0.1036078618111689], {6, 13} -> + Complex[0.017368709836144688`, 0.1497980533645717], {6, + 14} -> Complex[ + 0.13688251203715338`, 0.1036078618111689], {6, 15} -> + Complex[-0.09460977915945387, 0.11933166053315665`], {6, + 16} -> Complex[ + 0.02081012384838236, 0.14234469229568147`], {6, 17} -> + Complex[0.22516518597054483`, -0.1013577129442031], {6, + 18} -> Complex[-0.007671097858546656, \ +-0.0057340892009139326`], {6, 19} -> + Complex[0.21640487397840627`, 0.], {6, 20} -> + Complex[8.326672684688674*^-17, + 1.3877787807814457`*^-17], {6, 21} -> + Complex[0.030636763457055183`, -0.11355874224143539`], {6, + 22} -> + Complex[-0.029720518023860276`, -0.053544088868121745`], { + 6, 23} -> Complex[-0.06797212261961828, + 0.14596193524618625`], {6, 24} -> + Complex[-0.1599463611986579, 0.3186964622408383], {7, 8} -> + Complex[0.5226137655938699, 0.], {7, 9} -> + Complex[-0.08485061223426603, 0.11980477813807351`], {7, + 10} -> Complex[ + 0.17479296155780188`, 0.14732776197358494`], {7, 11} -> + Complex[0.11065480592987563`, -0.022498803837048064`], {7, + 12} -> + Complex[-0.13183382282185513`, -0.15967490110950291`], {7, + 13} -> Complex[ + 0.11065480592987563`, -0.022498803837048064`], {7, 14} -> + Complex[-0.13183382282185513`, -0.15967490110950291`], {7, + 15} -> Complex[-0.08485061223426603, + 0.11980477813807351`], {7, 16} -> + Complex[0.17479296155780188`, 0.14732776197358494`], {7, + 17} -> Complex[-1.7867651802561116`*^-16, + 1.1449174941446927`*^-16], {7, 18} -> + Complex[0.5226137655938699, 0.], {7, 19} -> + Complex[-0.007671097858546656, 0.0057340892009139326`], { + 7, 20} -> Complex[ + 0.2578656808252187, 0.10876030801336257`], {7, 21} -> + Complex[0.037739555243020126`, -0.012308050579545225`], { + 7, 22} -> + Complex[-0.13167457410588743`, -0.003054508743256395], {7, + 23} -> + Complex[-0.23097227051538433`, -0.42320695740719316`], {7, + 24} -> Complex[ + 0.29687889692943875`, 0.2375326567068352], {8, 9} -> + Complex[-0.17479296155780177`, 0.14732776197358502`], {8, + 10} -> Complex[ + 0.08485061223426596, 0.11980477813807357`], {8, 11} -> + Complex[-0.04249050665712762, -0.090268967382963], {8, + 12} -> Complex[ + 0.09710709374348331, 0.20541302638200343`], {8, 13} -> + Complex[-0.04249050665712762, -0.090268967382963], {8, + 14} -> Complex[ + 0.09710709374348331, 0.20541302638200343`], {8, 15} -> + Complex[-0.17479296155780177`, 0.14732776197358502`], {8, + 16} -> Complex[ + 0.08485061223426596, 0.11980477813807357`], {8, 17} -> + Complex[0.47738623440613004`, 0.], {8, 18} -> + Complex[-1.7867651802561116`*^-16, \ +-1.1449174941446927`*^-16], {8, 19} -> + Complex[0.22516518597054483`, 0.1013577129442031], {8, + 20} -> Complex[-0.17834040611966948`, + 0.12609948481270494`], {8, 21} -> + Complex[0.2616855917631509, -0.009667937264448562], {8, + 22} -> Complex[ + 0.053161902759652525`, -0.14451247653181823`], {8, 23} -> + Complex[-0.16418162796222252`, 0.13133580222113328`], {8, + 24} -> Complex[-0.42262163352679005`, + 0.31456841745062747`], {9, 10} -> + Complex[0.3800250972969439, 0.], {9, 11} -> + Complex[0.1899783589314758, -0.1365082746823964], {9, 12} -> + Complex[-0.15558917647409173`, -0.09979098803090053], {9, + 13} -> Complex[0.1899783589314758, -0.1365082746823964], { + 9, 14} -> + Complex[-0.15558917647409173`, -0.09979098803090053], {9, + 15} -> Complex[-8.673617379884035*^-17, \ +-8.326672684688674*^-17], {9, 16} -> Complex[0.3800250972969439, 0.], {9, 17} -> + Complex[0.08485061223426596, -0.11980477813807357`], {9, + 18} -> Complex[ + 0.17479296155780188`, -0.14732776197358494`], {9, 19} -> + Complex[0.02081012384838236, -0.14234469229568147`], {9, + 20} -> Complex[ + 0.13476314496413244`, -0.0767423516894553], {9, 21} -> + Complex[-0.15157725194981503`, 0.17844414940708186`], {9, + 22} -> Complex[-0.04368290380765367, \ +-0.027436459786564067`], {9, 23} -> + Complex[0.01747597846754094, 0.10203270358947614`], {9, + 24} -> Complex[ + 0.34087862056969404`, 0.1100388283919505], {10, 11} -> + Complex[-0.1905043599805688, -0.13762822610920328`], {10, + 12} -> Complex[ + 0.13800495411149877`, -0.3680867713233184], {10, 13} -> + Complex[-0.1905043599805688, -0.13762822610920328`], {10, + 14} -> Complex[ + 0.13800495411149877`, -0.3680867713233184], {10, 15} -> + Complex[0.619974902703056, 0.], {10, 16} -> + Complex[-8.673617379884035*^-17, + 8.326672684688674*^-17], {10, 17} -> + Complex[-0.17479296155780177`, -0.14732776197358502`], { + 10, 18} -> + Complex[-0.08485061223426603, -0.11980477813807351`], {10, + 19} -> + Complex[-0.09460977915945387, -0.11933166053315665`], {10, + 20} -> Complex[ + 0.12289396168675226`, 0.38209796458606343`], {10, 21} -> + Complex[-0.2346066684020043, -0.5335785389713812], {10, + 22} -> Complex[ + 0.08210404404737587, 0.05307307167276523], {10, 23} -> + Complex[-0.2917595352077467, 0.3784536838475334], {10, + 24} -> Complex[ + 0.33333532926317844`, 0.0052698415932301845`], {11, 12} -> + Complex[0.4724699088969763, 0.], {11, 13} -> + Complex[ + 1.8041124150158796`*^-16, -1.457167719820518*^-16], {11, + 14} -> Complex[0.4724699088969763, 0.], {11, 15} -> + Complex[0.13800495411149877`, 0.3680867713233184], {11, + 16} -> Complex[-0.15558917647409173`, + 0.09979098803090053], {11, 17} -> + Complex[0.09710709374348331, -0.20541302638200343`], {11, + 18} -> Complex[-0.13183382282185513`, + 0.15967490110950291`], {11, 19} -> + Complex[0.13688251203715338`, -0.1036078618111689], {11, + 20} -> Complex[-0.3239368353423375, + 0.27853835234466173`], {11, 21} -> + Complex[0.25833102475550235`, -0.37839808713437206`], {11, + 22} -> Complex[ + 0.01476498538421066, -0.0004105289599095289], {11, 23} -> + Complex[-0.0868392972934217, 0.014513008624828502`], {11, + 24} -> Complex[-0.16585075463443222`, + 0.4085141858532183], {12, 13} -> + Complex[0.5275300911030236, 0.], {12, 14} -> + Complex[1.8041124150158796`*^-16, + 1.457167719820518*^-16], {12, 15} -> + Complex[-0.1905043599805688, 0.13762822610920328`], {12, + 16} -> Complex[0.1899783589314758, 0.1365082746823964], { + 12, 17} -> + Complex[-0.04249050665712762, 0.090268967382963], {12, + 18} -> Complex[ + 0.11065480592987563`, 0.022498803837048064`], {12, 19} -> + Complex[0.017368709836144688`, -0.1497980533645717], {12, + 20} -> Complex[-0.19299966150123138`, \ +-0.3282026155242766], {12, 21} -> + Complex[-0.06250476454855013, 0.34969666557353624`], {12, + 22} -> Complex[-0.008146862343863803, + 0.10273781027166022`], {12, 23} -> + Complex[0.11666092373220538`, -0.218936695047531], {12, + 24} -> Complex[ + 0.161016937967343, 0.024863370726720582`], {13, 14} -> + Complex[0.4724699088969763, 0.], {13, 15} -> + Complex[0.13800495411149877`, 0.3680867713233184], {13, + 16} -> Complex[-0.15558917647409173`, + 0.09979098803090053], {13, 17} -> + Complex[0.09710709374348331, -0.20541302638200343`], {13, + 18} -> Complex[-0.13183382282185513`, + 0.15967490110950291`], {13, 19} -> + Complex[0.13688251203715338`, -0.1036078618111689], {13, + 20} -> Complex[-0.3239368353423375, + 0.27853835234466173`], {13, 21} -> + Complex[0.25833102475550235`, -0.37839808713437206`], {13, + 22} -> Complex[ + 0.01476498538421066, -0.0004105289599095289], {13, 23} -> + Complex[-0.0868392972934217, 0.014513008624828502`], {13, + 24} -> Complex[-0.16585075463443222`, + 0.4085141858532183], {14, 15} -> + Complex[-0.1905043599805688, 0.13762822610920328`], {14, + 16} -> Complex[0.1899783589314758, 0.1365082746823964], { + 14, 17} -> + Complex[-0.04249050665712762, 0.090268967382963], {14, + 18} -> Complex[ + 0.11065480592987563`, 0.022498803837048064`], {14, 19} -> + Complex[0.017368709836144688`, -0.1497980533645717], {14, + 20} -> Complex[-0.19299966150123138`, \ +-0.3282026155242766], {14, 21} -> + Complex[-0.06250476454855013, 0.34969666557353624`], {14, + 22} -> Complex[-0.008146862343863803, + 0.10273781027166022`], {14, 23} -> + Complex[0.11666092373220538`, -0.218936695047531], {14, + 24} -> Complex[ + 0.161016937967343, 0.024863370726720582`], {15, 16} -> + Complex[0.3800250972969439, 0.], {15, 17} -> + Complex[0.08485061223426596, -0.11980477813807357`], {15, + 18} -> Complex[ + 0.17479296155780188`, -0.14732776197358494`], {15, 19} -> + Complex[0.02081012384838236, -0.14234469229568147`], {15, + 20} -> Complex[ + 0.13476314496413244`, -0.0767423516894553], {15, 21} -> + Complex[-0.15157725194981503`, 0.17844414940708186`], {15, + 22} -> + Complex[-0.04368290380765367, -0.027436459786564067`], { + 15, 23} -> + Complex[0.01747597846754094, 0.10203270358947614`], {15, + 24} -> Complex[ + 0.34087862056969404`, 0.1100388283919505], {16, 17} -> + Complex[-0.17479296155780177`, -0.14732776197358502`], { + 16, 18} -> + Complex[-0.08485061223426603, -0.11980477813807351`], {16, + 19} -> + Complex[-0.09460977915945387, -0.11933166053315665`], {16, + 20} -> Complex[ + 0.12289396168675226`, 0.38209796458606343`], {16, 21} -> + Complex[-0.2346066684020043, -0.5335785389713812], {16, + 22} -> Complex[ + 0.08210404404737587, 0.05307307167276523], {16, 23} -> + Complex[-0.2917595352077467, 0.3784536838475334], {16, + 24} -> Complex[ + 0.33333532926317844`, 0.0052698415932301845`], {17, 18} -> + Complex[0.5226137655938699, 0.], {17, 19} -> + Complex[-0.007671097858546656, 0.0057340892009139326`], { + 17, 20} -> + Complex[0.2578656808252187, 0.10876030801336257`], {17, + 21} -> Complex[ + 0.037739555243020126`, -0.012308050579545225`], {17, 22} -> + Complex[-0.13167457410588743`, -0.003054508743256395], { + 17, 23} -> + Complex[-0.23097227051538433`, -0.42320695740719316`], { + 17, 24} -> + Complex[0.29687889692943875`, 0.2375326567068352], {18, + 19} -> Complex[ + 0.22516518597054483`, 0.1013577129442031], {18, 20} -> + Complex[-0.17834040611966948`, 0.12609948481270494`], {18, 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NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 795724, 15394] -NotebookOptionsPosition[ 782483, 15135] -NotebookOutlinePosition[ 786311, 15213] -CellTagsIndexPosition[ 786185, 15207] +NotebookDataLength[ 508091, 10122] +NotebookOptionsPosition[ 495560, 9873] +NotebookOutlinePosition[ 499168, 9946] +CellTagsIndexPosition[ 499081, 9941] WindowTitle->WickState WindowFrame->Normal*) @@ -27,11 +27,11 @@ Cell[BoxData[GridBox[{ TemplateBox[{12}, "Spacer1"], Cell["Q3 SYMBOL", "PacletNameCell", TextAlignment->Center,ExpressionUUID-> - "ed6a07e9-08c8-44bc-aba0-ade2fc067969"], + "642d4614-2dcc-4568-bf4b-c0f0872df56f"], TemplateBox[{8}, "Spacer1"]}]], TextAlignment->Center,ExpressionUUID-> - "6b191758-26bf-4c86-9cfa-203141c1ff40"], + "71acb690-5e44-452b-b613-36401d8e7227"], Background->RGBColor[0.490196, 0.576471, 0.690196], ItemSize->Full], ""} }, @@ -54,23 +54,25 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "c2146488-2a74-4f8b-a93c-a5e27f1df115"] - }],ExpressionUUID->"6caf5588-a0cf-4e00-b63d-67f564dd0fa1"], + "b97f504f-ad7c-4db3-989c-e1f3ba1fed6b"] + }],ExpressionUUID->"f1cec2a8-df4b-4b40-94e7-bd3ac851b3cf"], StripOnInput->False],{ "\"WickExpectation\"" :> Documentation`HelpLookup["paclet:Q3/ref/WickExpectation"], "\"WickOperator\"" :> Documentation`HelpLookup["paclet:Q3/ref/WickOperator"], - "\"Fermion\"" :> Documentation`HelpLookup["paclet:Q3/ref/Fermion"], - "\"Majorana\"" :> Documentation`HelpLookup["paclet:Q3/ref/Majorana"], - "\"Measurement\"" :> - Documentation`HelpLookup["paclet:Q3/ref/Measurement"]}, + "\"WickUnitary\"" :> + Documentation`HelpLookup["paclet:Q3/ref/WickUnitary"], + "\"Measurement\"" :> + Documentation`HelpLookup["paclet:Q3/ref/Measurement"], "\"Fermion\"" :> + Documentation`HelpLookup["paclet:Q3/ref/Fermion"], "\"Majorana\"" :> + Documentation`HelpLookup["paclet:Q3/ref/Majorana"]}, Appearance->None, MenuAppearance->Automatic, MenuStyle->"SeeAlso"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "1475884a-f3a6-40ca-bc82-d1e9e98119a4"], + "66b6f6d5-4680-4307-a943-c109b83d742c"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -89,8 +91,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "ef66d311-5a2c-4019-8916-2aeb495ded2e"] - }],ExpressionUUID->"626bdf73-5a0f-4e05-bdbd-e536268e3801"], + "057fef85-d85f-4b73-8b3d-c9aa43e6c952"] + }],ExpressionUUID->"e632f5ed-46f6-44f1-b164-e668cec4b082"], StripOnInput->False],{ "\"Quantum Many-Body Systems\"" :> Documentation`HelpLookup["paclet:Q3/guide/QuantumManyBodySystems"], @@ -100,7 +102,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"MoreAbout"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "893f4ecd-263e-4266-af05-c5ab91164e11"], + 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InterpretationBox[ + SubscriptBox["c", - Complex[0.000016605652687226302`, 0.0991690796661601]}, { + TemplateBox[{",", "\",\"", "2", "3", "0"}, + "RowWithSeparators"]], + $CellContext`c[2, 3, 0]], ",", + InterpretationBox[ + SubscriptBox["c", - Complex[-9.943302374432365*^-21, \ --3.3214464132085736`*^-7], - Complex[-1.1079404291781963`*^-8, 0.0003316699566825007], + TemplateBox[{",", "\",\"", "2", "2", "1"}, + "RowWithSeparators"]], + $CellContext`c[2, 2, 1]], ",", + InterpretationBox[ + SubscriptBox["c", - Complex[0.000016605652687224523`, -0.0991690796661601], - Complex[-0.00995010820050242, 2.083853062282483*^-19]}}, - Selectable -> False, Editable -> False, - SelectWithContents -> True]}], "}"}], ",", - RowBox[{"{", - RowBox[{ + TemplateBox[{",", "\",\"", "1", "1", "1"}, + "RowWithSeparators"]], + $CellContext`c[1, 1, 1]], ",", InterpretationBox[ - RowBox[{ - TagBox["MatrixObject", "SummaryHead"], "[", + SubscriptBox["c", - DynamicModuleBox[{Typeset`open$$ = False, - 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AbsoluteCurrentValue[Magnification])}]], + Dynamic[{Automatic, + 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ + Magnification])}]], GridBox[{{ RowBox[{ - TagBox["\"Type: \"", "SummaryItemAnnotation"], + TagBox["\"Transforms: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["\"Dense\"", "SummaryItem"]}]}, { + TagBox["3", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Dimensions: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox[ RowBox[{"{", RowBox[{"4", ",", "4"}], "}"}], "SummaryItem"]}]}}, - GridBoxAlignment -> {"Columns" -> {{Left}}, - "Rows" -> {{Automatic}}}, AutoDelete -> False, - GridBoxItemSize -> {"Columns" -> {{Automatic}}, - "Rows" -> {{Automatic}}}, GridBoxSpacings -> { + GridBoxAlignment -> { + "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, + AutoDelete -> False, + GridBoxItemSize -> { + "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, + GridBoxSpacings -> { "Columns" -> {{2}}, "Rows" -> {{Automatic}}}, - BaseStyle -> {ShowStringCharacters 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Automatic, Method -> "Preemptive"], Alignment -> {Center, Center}, ImageSize -> - Dynamic[{Automatic, 3.5 (CurrentValue["FontCapHeight"]/ - AbsoluteCurrentValue[Magnification])}]], + Dynamic[{Automatic, + 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ + Magnification])}]], GridBox[{{ RowBox[{ - TagBox["\"Type: \"", "SummaryItemAnnotation"], + TagBox["\"Transforms: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["\"Dense\"", "SummaryItem"]}]}, { + TagBox["3", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Dimensions: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", @@ -3551,124 +2385,16 @@ HACo2sx4 RowBox[{"{", RowBox[{"4", ",", "4"}], "}"}], "SummaryItem"]}]}, { RowBox[{ - TagBox["\"Elements: \"", "SummaryItemAnnotation"], + TagBox["\"Blocks: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox[ - TagBox[ - RowBox[{"(", "\[NoBreak]", - GridBox[{{ - RowBox[{ - RowBox[{"0.9411238678566561`", "\[VeryThinSpace]"}], "+", - - RowBox[{"0.19341019859437134`", " ", 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DynamicModuleBox[{Typeset`open$$ = False, Typeset`embedState$$ = "Ready"}, @@ -3684,11 +2410,39 @@ HACo2sx4 "Preemptive"], Alignment -> {Center, Center}, ImageSize -> Dynamic[{Automatic, 3.5 (CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification])}]], + GraphicsBox[ + RasterBox[CompressedData[" +1:eJxTTMoPSmVmYGBggWIQ26p8d9wspXP2c4wXbTH/ccX+xSLFx4GrXtmTKr6z +28B155YX9jfcl6+rmfbW/mRE7r/4r+9HzSEgDgDactH8 + "], {{0, 0}, {4, 4}}, {0, 1}], AspectRatio -> 1, Axes -> + False, AxesStyle -> {}, Frame -> False, + FrameLabel -> {None, None}, FrameStyle -> Directive[ + Opacity[0.5], + Thickness[Tiny], + RGBColor[0.368417, 0.506779, 0.709798]], + FrameTicks -> {{None, None}, {None, None}}, + GridLinesStyle -> Directive[ + GrayLevel[0.5, 0.4]], ImageSize -> Dynamic[{ + Automatic, 3.5 (CurrentValue["FontCapHeight"]/ + AbsoluteCurrentValue[Magnification])}], + Method -> {"GridLinesInFront" -> True, "AxisPadding" -> + Scaled[0.02], "DefaultBoundaryStyle" -> Automatic, + "DefaultGraphicsInteraction" -> {"Version" -> 1.2, + "TrackMousePosition" -> {True, False}, "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> { + "ratio" -> 2}, "Droplines" -> {"freeformCursorMode" -> + True, "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultPlotStyle" -> Automatic, "DomainPadding" -> + Scaled[0.02], "RangePadding" -> Scaled[0.05]}, + PlotRangePadding -> Automatic, AxesStyle -> Large, + FrameStyle -> Large, ImageSize -> Large, PlotRangePadding -> + Scaled[0.025]], GridBox[{{ RowBox[{ - TagBox["\"Type: \"", "SummaryItemAnnotation"], + TagBox[ + "\"Specified elements: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["\"Dense\"", "SummaryItem"]}]}, { + TagBox["4", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Dimensions: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", @@ -3717,11 +2471,39 @@ HACo2sx4 "Preemptive"], Alignment -> {Center, Center}, ImageSize -> Dynamic[{Automatic, 3.5 (CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification])}]], + GraphicsBox[ + RasterBox[CompressedData[" +1:eJxTTMoPSmVmYGBggWIQ26p8d9wspXP2c4wXbTH/ccX+xSLFx4GrXtmTKr6z +28B155YX9jfcl6+rmfbW/mRE7r/4r+9HzSEgDgDactH8 + "], {{0, 0}, {4, 4}}, {0, 1}], AspectRatio -> 1, Axes -> + False, AxesStyle -> {}, Frame -> False, + FrameLabel -> {None, None}, FrameStyle -> Directive[ + Opacity[0.5], + Thickness[Tiny], + RGBColor[0.368417, 0.506779, 0.709798]], + FrameTicks -> {{None, None}, {None, None}}, + GridLinesStyle -> Directive[ + GrayLevel[0.5, 0.4]], ImageSize -> + Dynamic[{Automatic, 3.5 (CurrentValue["FontCapHeight"]/ + AbsoluteCurrentValue[Magnification])}], + Method -> {"GridLinesInFront" -> True, "AxisPadding" -> + Scaled[0.02], "DefaultBoundaryStyle" -> Automatic, + "DefaultGraphicsInteraction" -> {"Version" -> 1.2, + "TrackMousePosition" -> {True, False}, "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> { + "ratio" -> 2}, "Droplines" -> {"freeformCursorMode" -> + True, "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultPlotStyle" -> Automatic, "DomainPadding" -> + Scaled[0.02], "RangePadding" -> Scaled[0.05]}, + PlotRangePadding -> Automatic, AxesStyle -> Large, + FrameStyle -> Large, ImageSize -> Large, PlotRangePadding -> + Scaled[0.025]], GridBox[{{ RowBox[{ - TagBox["\"Type: \"", "SummaryItemAnnotation"], + TagBox[ + "\"Specified elements: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["\"Dense\"", "SummaryItem"]}]}, { + TagBox["4", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Dimensions: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", @@ -3729,107 +2511,79 @@ HACo2sx4 RowBox[{"{", RowBox[{"4", ",", "4"}], "}"}], "SummaryItem"]}]}, { RowBox[{ - TagBox["\"Elements: \"", "SummaryItemAnnotation"], + TagBox["\"Default: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox[ - TagBox[ - RowBox[{"(", "\[NoBreak]", - GridBox[{{"0.03920689940953653`", - RowBox[{ - RowBox[{"-", "0.00026278035189146974`"}], "+", - RowBox[{"0.1934101985943713`", 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"\[ImaginaryI]"}]}], - RowBox[{"-", "7.030241935355314`*^-7"}], + TagBox["\"Elements:\"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox["\"\"", "SummaryItem"]}]}, { + TagBox[ + TagBox[ + GridBox[{{ RowBox[{ - RowBox[{"0.000262780351891468`", "\[VeryThinSpace]"}], - "+", - RowBox[{"0.19341019859437125`", " ", - "\[ImaginaryI]"}]}]}, { + RowBox[{"{", + RowBox[{"1", ",", "1"}], "}"}], "\[Rule]", + StyleBox[ + PaneBox[ + "1", ContentPadding -> False, FrameMargins -> 0, + StripOnInput -> True, BaselinePosition -> Baseline, + ImageSize -> {{1, 300}, Automatic}], LineBreakWithin -> + False]}]}, { RowBox[{ - RowBox[{"0.`", "\[VeryThinSpace]"}], "-", - - RowBox[{"0.000010515196858957876`", " ", - "\[ImaginaryI]"}]}], + RowBox[{"{", + RowBox[{"2", ",", "2"}], "}"}], "\[Rule]", + StyleBox[ + PaneBox[ + "1", ContentPadding -> False, FrameMargins -> 0, + StripOnInput -> True, BaselinePosition -> Baseline, + ImageSize -> {{1, 300}, Automatic}], LineBreakWithin -> + False]}]}, { RowBox[{ 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Scaled[0.05]}, + PlotRangePadding -> Automatic, AxesStyle -> Large, + FrameStyle -> Large, ImageSize -> Large, PlotRangePadding -> + Scaled[0.025]], GridBox[{{ RowBox[{ - TagBox["\"Type: \"", "SummaryItemAnnotation"], + TagBox[ + "\"Specified elements: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["\"Dense\"", "SummaryItem"]}]}, { + TagBox["0", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Dimensions: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", @@ -3878,7 +2660,128 @@ HACo2sx4 "Preemptive"], Alignment -> {Center, Center}, ImageSize -> Dynamic[{Automatic, 3.5 (CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification])}]], - GridBox[{{ + GraphicsBox[ + RasterBox[CompressedData[" +1:eJxTTMoPSmVmYGBggWIQ26p8d9wspXP2c4wXbTH/ccX+xSLFx4GrXtmPitNH +HACo2sx4 + "], {{0, 0}, {4, 4}}, {0, 1}], AspectRatio -> 1, Axes -> + False, AxesStyle -> {}, Frame -> False, + FrameLabel -> {None, None}, FrameStyle -> Directive[ + Opacity[0.5], + Thickness[Tiny], + RGBColor[0.368417, 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+ "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, + AutoDelete -> False, + GridBoxItemSize -> { + "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, + GridBoxSpacings -> { "Columns" -> {{2}}, "Rows" -> {{Automatic}}}, - BaseStyle -> {ShowStringCharacters -> False, NumberMarks -> - False, PrintPrecision -> 3, ShowSyntaxStyles -> False}]}}, - GridBoxAlignment -> {"Columns" -> {{Left}}, - "Rows" -> {{Top}}}, AutoDelete -> False, - GridBoxItemSize -> {"Columns" -> {{Automatic}}, - "Rows" -> {{Automatic}}}, BaselinePosition -> {1, 1}], - True -> GridBox[{{ + BaseStyle -> { + ShowStringCharacters -> False, NumberMarks -> False, + PrintPrecision -> 3, ShowSyntaxStyles -> False}]}}, + GridBoxAlignment -> { + "Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> + False, GridBoxItemSize -> { + "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, + BaselinePosition -> {1, 1}], True -> GridBox[{{ PaneBox[ ButtonBox[ DynamicBox[ FEPrivate`FrontEndResource[ - "FEBitmaps", "SummaryBoxCloser"]], ButtonFunction :> ( - Typeset`open$$ = False), Appearance -> None, - BaseStyle -> {}, Evaluator -> Automatic, Method -> + "FEBitmaps", "SummaryBoxCloser"]], + ButtonFunction :> (Typeset`open$$ = False), Appearance -> + None, BaseStyle -> {}, Evaluator -> Automatic, Method -> "Preemptive"], Alignment -> {Center, Center}, ImageSize -> - Dynamic[{Automatic, 3.5 (CurrentValue["FontCapHeight"]/ - AbsoluteCurrentValue[Magnification])}]], - GraphicsBox[ - RasterBox[CompressedData[" -1:eJxTTMoPSmVmYGBggWIQ26p8d9wspXP2c4wXbTH/ccX+xSLFx4GrXtmPitNH -HACo2sx4 - "], {{0, 0}, {4, 4}}, {0, 1}], AspectRatio -> 1, Axes -> - False, AxesStyle -> {}, Frame -> False, - FrameLabel -> {None, None}, FrameStyle -> Directive[ - Opacity[0.5], - Thickness[Tiny], - RGBColor[0.368417, 0.506779, 0.709798]], - FrameTicks -> {{None, None}, {None, None}}, - GridLinesStyle -> Directive[ - GrayLevel[0.5, 0.4]], ImageSize -> - Dynamic[{Automatic, 3.5 (CurrentValue["FontCapHeight"]/ - AbsoluteCurrentValue[Magnification])}], - Method -> {"GridLinesInFront" -> True, "AxisPadding" -> - Scaled[0.02], "DefaultBoundaryStyle" -> Automatic, - "DefaultGraphicsInteraction" -> {"Version" -> 1.2, - "TrackMousePosition" -> {True, False}, "Effects" -> { - "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> { - "ratio" -> 2}, "Droplines" -> {"freeformCursorMode" -> - True, "placement" -> {"x" -> "All", "y" -> "None"}}}}, - "DefaultPlotStyle" -> Automatic, "DomainPadding" -> - Scaled[0.02], "RangePadding" -> Scaled[0.05]}, - PlotRangePadding -> Automatic, AxesStyle -> Large, - FrameStyle -> Large, ImageSize -> Large, PlotRangePadding -> - Scaled[0.025]], + Dynamic[{Automatic, + 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ + Magnification])}]], GridBox[{{ RowBox[{ - TagBox[ - "\"Specified elements: \"", "SummaryItemAnnotation"], + TagBox["\"Transforms: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["0", "SummaryItem"]}]}, { + TagBox["3", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Dimensions: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", @@ -9326,39 +5177,16 @@ HACo2sx4 RowBox[{"{", RowBox[{"4", ",", "4"}], "}"}], "SummaryItem"]}]}, { RowBox[{ - TagBox["\"Default: \"", "SummaryItemAnnotation"], - "\[InvisibleSpace]", - TagBox["0", "SummaryItem"]}]}, { - RowBox[{ - TagBox["\"Density: \"", "SummaryItemAnnotation"], + TagBox["\"Blocks: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["0.`", "SummaryItem"]}]}, { + TagBox[ + RowBox[{"{", RowBox[{ - TagBox["\"Elements: \"", "SummaryItemAnnotation"], - "\[InvisibleSpace]", - TagBox["None", "SummaryItem"]}]}}, GridBoxAlignment -> { - "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, - AutoDelete -> False, GridBoxItemSize -> { - "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, - GridBoxSpacings -> {"Columns" -> {{2}}, - "Rows" -> {{Automatic}}}, BaseStyle -> { - ShowStringCharacters -> False, NumberMarks -> False, - PrintPrecision -> 3, ShowSyntaxStyles -> False}]}}, - GridBoxAlignment -> {"Columns" -> {{Left}}, - "Rows" -> {{Top}}}, AutoDelete -> False, - GridBoxItemSize -> {"Columns" -> {{Automatic}}, - "Rows" -> {{Automatic}}}, BaselinePosition -> {1, 1}]}, - Dynamic[Typeset`open$$], ImageSize -> Automatic]}, - "SummaryPanel"], DynamicModuleValues :> {}], "]"}], - SparseArray[Automatic, {4, 4}, 0, { - 1, {{0, 0, 0, 0, 0}, {}}, {}}], Selectable -> False, - Editable -> False, SelectWithContents -> True]}], "}"}], - ",", RowBox[{"{", RowBox[{ InterpretationBox[ RowBox[{ - TagBox["MatrixObject", "SummaryHead"], "[", + TagBox["SparseArray", "SummaryHead"], "[", DynamicModuleBox[{Typeset`open$$ = False, Typeset`embedState$$ = "Ready"}, @@ -9373,12 +5201,40 @@ HACo2sx4 BaseStyle -> {}, Evaluator -> Automatic, Method -> "Preemptive"], Alignment -> {Center, Center}, ImageSize -> Dynamic[{Automatic, 3.5 (CurrentValue["FontCapHeight"]/ - AbsoluteCurrentValue[Magnification])}]], + AbsoluteCurrentValue[Magnification])}]], + GraphicsBox[ + RasterBox[CompressedData[" +1:eJxTTMoPSmVmYGBggWIQ26p8d9wspXP2c4wXbTH/ccX+xSLFx4GrXtmTKr6z +28B155YX9jfcl6+rmfbW/mRE7r/4r+9HzSEgDgDactH8 + "], {{0, 0}, {4, 4}}, {0, 1}], AspectRatio -> 1, Axes -> + False, AxesStyle -> {}, Frame -> False, + FrameLabel -> {None, None}, FrameStyle -> Directive[ + Opacity[0.5], + Thickness[Tiny], + RGBColor[0.368417, 0.506779, 0.709798]], + FrameTicks -> {{None, None}, {None, None}}, + GridLinesStyle -> Directive[ + GrayLevel[0.5, 0.4]], ImageSize -> + Dynamic[{Automatic, 3.5 (CurrentValue["FontCapHeight"]/ + AbsoluteCurrentValue[Magnification])}], + Method -> {"GridLinesInFront" -> True, "AxisPadding" -> + Scaled[0.02], "DefaultBoundaryStyle" -> Automatic, + "DefaultGraphicsInteraction" -> {"Version" -> 1.2, + "TrackMousePosition" -> {True, False}, "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> { + "ratio" -> 2}, "Droplines" -> {"freeformCursorMode" -> + True, "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultPlotStyle" -> Automatic, "DomainPadding" -> + Scaled[0.02], "RangePadding" -> Scaled[0.05]}, + PlotRangePadding -> Automatic, AxesStyle -> Large, + FrameStyle -> Large, ImageSize -> Large, PlotRangePadding -> + Scaled[0.025]], GridBox[{{ RowBox[{ - TagBox["\"Type: \"", "SummaryItemAnnotation"], + TagBox[ + "\"Specified elements: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["\"Dense\"", "SummaryItem"]}]}, { + TagBox["4", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Dimensions: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", @@ -9407,11 +5263,39 @@ HACo2sx4 "Preemptive"], Alignment -> {Center, Center}, ImageSize -> Dynamic[{Automatic, 3.5 (CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification])}]], + GraphicsBox[ + RasterBox[CompressedData[" +1:eJxTTMoPSmVmYGBggWIQ26p8d9wspXP2c4wXbTH/ccX+xSLFx4GrXtmTKr6z +28B155YX9jfcl6+rmfbW/mRE7r/4r+9HzSEgDgDactH8 + "], {{0, 0}, {4, 4}}, {0, 1}], AspectRatio -> 1, Axes -> + False, AxesStyle -> {}, Frame -> False, + FrameLabel -> {None, None}, FrameStyle -> Directive[ + Opacity[0.5], + Thickness[Tiny], + RGBColor[0.368417, 0.506779, 0.709798]], + FrameTicks -> {{None, None}, {None, None}}, + GridLinesStyle -> Directive[ + GrayLevel[0.5, 0.4]], ImageSize -> + Dynamic[{Automatic, 3.5 (CurrentValue["FontCapHeight"]/ + AbsoluteCurrentValue[Magnification])}], + Method -> {"GridLinesInFront" -> True, "AxisPadding" -> + Scaled[0.02], "DefaultBoundaryStyle" -> Automatic, + "DefaultGraphicsInteraction" -> {"Version" -> 1.2, + "TrackMousePosition" -> {True, False}, "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> { + "ratio" -> 2}, "Droplines" -> {"freeformCursorMode" -> + True, "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultPlotStyle" -> Automatic, "DomainPadding" -> + Scaled[0.02], "RangePadding" -> Scaled[0.05]}, + PlotRangePadding -> Automatic, AxesStyle -> Large, + FrameStyle -> Large, ImageSize -> Large, PlotRangePadding -> + Scaled[0.025]], GridBox[{{ RowBox[{ - TagBox["\"Type: \"", 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GraphicsBox[ + RasterBox[CompressedData[" +1:eJxTTMoPSmVmYGBggWIQ26p8d9wspXP2c4wXbTH/ccX+xSLFx4GrXtmPitNH +HACo2sx4 + "], {{0, 0}, {4, 4}}, {0, 1}], AspectRatio -> 1, Axes -> + False, AxesStyle -> {}, Frame -> False, + FrameLabel -> {None, None}, FrameStyle -> Directive[ + Opacity[0.5], + Thickness[Tiny], + RGBColor[0.368417, 0.506779, 0.709798]], + FrameTicks -> {{None, None}, {None, None}}, + GridLinesStyle -> Directive[ + GrayLevel[0.5, 0.4]], ImageSize -> + Dynamic[{Automatic, 3.5 (CurrentValue["FontCapHeight"]/ + AbsoluteCurrentValue[Magnification])}], + Method -> {"GridLinesInFront" -> True, "AxisPadding" -> + Scaled[0.02], "DefaultBoundaryStyle" -> Automatic, + "DefaultGraphicsInteraction" -> {"Version" -> 1.2, + "TrackMousePosition" -> {True, False}, "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> { + "ratio" -> 2}, "Droplines" -> {"freeformCursorMode" -> + True, "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultPlotStyle" -> Automatic, "DomainPadding" -> + Scaled[0.02], "RangePadding" -> Scaled[0.05]}, + PlotRangePadding -> Automatic, AxesStyle -> Large, + FrameStyle -> Large, ImageSize -> Large, PlotRangePadding -> + Scaled[0.025]], GridBox[{{ RowBox[{ - TagBox["\"Type: \"", "SummaryItemAnnotation"], + TagBox[ + "\"Specified elements: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["\"Dense\"", "SummaryItem"]}]}, { + TagBox["0", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Dimensions: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", @@ -9589,82 +5452,57 @@ HACo2sx4 "Preemptive"], Alignment -> {Center, Center}, ImageSize -> Dynamic[{Automatic, 3.5 (CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification])}]], - GridBox[{{ - RowBox[{ - TagBox["\"Type: \"", "SummaryItemAnnotation"], - "\[InvisibleSpace]", - TagBox["\"Dense\"", "SummaryItem"]}]}, { - RowBox[{ - TagBox["\"Dimensions: \"", "SummaryItemAnnotation"], - "\[InvisibleSpace]", - TagBox[ - RowBox[{"{", - RowBox[{"4", ",", "4"}], "}"}], 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False]}]}, { + RowBox[{"0.0003106339104423192`", " ", + "\[ImaginaryI]"}]}], ContentPadding -> False, + FrameMargins -> 0, StripOnInput -> True, BaselinePosition -> + Baseline, ImageSize -> {{1, 300}, Automatic}], + LineBreakWithin -> False]}]}, { RowBox[{ RowBox[{"{", RowBox[{"1", ",", "4"}], "}"}], "\[Rule]", StyleBox[ PaneBox[ RowBox[{ - RowBox[{"-", "1.107940429399415`*^-8"}], "-", - - RowBox[{"2.241454781417565`*^-19", " ", - "\[ImaginaryI]"}]}], ContentPadding -> False, - FrameMargins -> 0, StripOnInput -> True, BaselinePosition -> - Baseline, ImageSize -> {{1, 300}, Automatic}], - LineBreakWithin -> False]}]}, { + RowBox[{"0.32171504751621155`", "\[VeryThinSpace]"}], "-", + RowBox[{"0.15725504484569033`", " ", "\[ImaginaryI]"}]}], + ContentPadding -> False, FrameMargins -> 0, StripOnInput -> + True, BaselinePosition -> Baseline, + ImageSize -> {{1, 300}, Automatic}], LineBreakWithin -> + False]}]}, { RowBox[{ RowBox[{"{", RowBox[{"1", ",", "5"}], "}"}], "\[Rule]", StyleBox[ 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0.004784170182671579, -2.7273105648872874`*^-16], {10, - 13} -> Complex[-0.07261962399452816, - 0.21892155188592985`], {10, 14} -> - Complex[ - 0.004784170182671579, -2.7273105648872874`*^-16], {10, - 15} -> Complex[0.8452681141734937, 0.], {10, 16} -> - Complex[ - 4.163336342344337*^-17, -6.938893903907228*^-18], {10, - 17} -> Complex[ - 0.05379673174522864, -0.0007416401460346826], {10, 18} -> - Complex[-0.051397296777308234`, -0.01859492038010707], { - 10, 19} -> - Complex[-0.03773674551714698, 0.19081560591987778`], {10, - 20} -> Complex[-0.026634445818647372`, - 0.0778365212048926], {10, 21} -> - Complex[0.8470224364690501, 0.1883181155038509], {10, 22} -> - Complex[ - 0.000016933675279148797`, -0.00012383773397451302`], {10, - 23} -> Complex[ - 0.0012981885381123139`, -0.008523700770098444], {10, 24} -> - Complex[0.7859004070182107, 0.2609534572619835], {11, 12} -> - Complex[0.8452681141734925, 0.], {11, 13} -> - Complex[-2.901596064114331*^-17, \ --1.0211829212097848`*^-17], {11, 14} -> - Complex[0.8452681141734925, 0.], {11, 15} -> - Complex[0.004784170182671579, 2.7273105648872874`*^-16], { - 11, 16} -> - Complex[0.07261962399452813, -0.21892155188592954`], {11, - 17} -> Complex[-0.00915537197655327, - 0.10974136766841047`], {11, 18} -> - Complex[-0.05329448030995345, 0.16271619670523418`], {11, - 19} -> Complex[ - 0.8470224364690492, 0.18831811550385083`], {11, 20} -> - Complex[-0.00001693367527912708, 0.0001238377339745108], { - 11, 21} -> - Complex[-0.03773674551714702, 0.190815605919878], {11, - 22} -> Complex[ - 0.026634445818647358`, -0.0778365212048925], {11, 23} -> - Complex[-0.08472202645669408, 0.26947715803208194`], {11, - 24} -> Complex[-0.08342383791858181, - 0.26963368038096003`], {12, 13} -> - Complex[0.15473188582650818`, 0.], {12, 14} -> - Complex[-2.901596064114331*^-17, - 1.0211829212097848`*^-17], {12, 15} -> - Complex[-0.07261962399452816, -0.21892155188592985`], {12, - 16} -> Complex[-0.004784170182671576, - 2.079127072330406*^-17], {12, 17} -> - Complex[0.05327787465726625, 0.2618852763713944], {12, - 18} -> Complex[-0.0007781305712618784, - 0.01090395795893278], {12, 19} -> - Complex[ - 0.000017636699472639164`, 0.0001238377339745129], {12, - 20} -> Complex[ - 0.05515660930576871, 0.00012490783968316282`], {12, 21} -> - Complex[-0.026634445818647365`, -0.271246719799264], {12, - 22} -> Complex[-0.0009452962128001206, \ --1.8498670906598045`*^-6], {12, 23} -> - Complex[-0.0013060825544481437`, 0.27807929010804266`], { - 12, 24} -> - Complex[6.938893903907228*^-18, -0.27807929010804266`], { - 13, 14} -> Complex[0.8452681141734925, 0.], {13, 15} -> - Complex[0.004784170182671579, 2.7273105648872874`*^-16], { - 13, 16} -> - Complex[0.07261962399452813, -0.21892155188592954`], {13, - 17} -> Complex[-0.00915537197655327, - 0.10974136766841047`], {13, 18} -> - Complex[-0.05329448030995345, 0.16271619670523418`], {13, + 0.24953605886718944`, -0.23726268416179194`], {6, 13} -> + Complex[-0.02984323129946001, -0.01090215294864004], {6, + 14} -> Complex[ + 0.10286685511517638`, 0.0766374954583717], {6, 15} -> + Complex[0.3009141283362328, 0.], {6, 16} -> + Complex[-0.04975850381773862, 0.06377388833703419], {6, + 17} -> Complex[-0.15900572219376757`, + 0.05750044204194605], {6, 18} -> + Complex[0.30539523194520163`, 0.10550264339992507`], {6, 19} -> Complex[ - 0.8470224364690492, 0.18831811550385083`], {13, 20} -> - Complex[-0.00001693367527912708, 0.0001238377339745108], { - 13, 21} -> - Complex[-0.03773674551714702, 0.190815605919878], {13, - 22} -> Complex[ - 0.026634445818647358`, -0.0778365212048925], {13, 23} -> - Complex[-0.08472202645669408, 0.26947715803208194`], {13, - 24} -> Complex[-0.08342383791858181, - 0.26963368038096003`], {14, 15} -> - Complex[-0.07261962399452816, -0.21892155188592985`], {14, - 16} -> Complex[-0.004784170182671576, - 2.079127072330406*^-17], {14, 17} -> - Complex[0.05327787465726625, 0.2618852763713944], {14, - 18} -> Complex[-0.0007781305712618784, - 0.01090395795893278], {14, 19} -> - Complex[ - 0.000017636699472639164`, 0.0001238377339745129], {14, + 0.3910495342752539, -0.025967555373879958`], {6, 20} -> + Complex[-0.0981351838004039, -0.2911740023151237], {7, 8} -> + Complex[0.4155199590831424, 0.], {7, 9} -> + Complex[0.04268060967745012, 0.023059250426567188`], {7, + 10} -> Complex[-0.10752531345447021`, \ +-0.09271398750024093], {7, 11} -> + Complex[0.04268060967745012, 0.023059250426567188`], {7, + 12} -> Complex[-0.10752531345447021`, \ +-0.09271398750024093], {7, 13} -> + Complex[-6.938893903907228*^-18, \ +-2.7755575615628914`*^-17], {7, 14} -> + Complex[0.4155199590831424, 0.], {7, 15} -> + Complex[0.10286685511517638`, -0.0766374954583717], {7, + 16} -> Complex[ + 0.15338653623116225`, -0.06791293700745762], {7, 17} -> + Complex[-0.3323619403479582, 0.050870880986855216`], {7, + 18} -> Complex[ + 0.06560512137155002, 0.16228715013884426`], {7, 19} -> + Complex[0.05020845049867415, -0.26771648586524205`], {7, + 20} -> Complex[-0.021712677276518863`, + 0.058041075609157985`], {8, 9} -> + Complex[0.10752531345447036`, -0.09271398750024114], {8, + 10} -> Complex[-0.04268060967745001, + 0.023059250426567146`], {8, 11} -> + Complex[0.10752531345447036`, -0.09271398750024114], {8, + 12} -> Complex[-0.04268060967745001, + 0.023059250426567146`], {8, 13} -> + Complex[0.5844800409168576, 0.], {8, 14} -> + Complex[-6.938893903907228*^-18, + 2.7755575615628914`*^-17], {8, 15} -> + Complex[-0.02984323129946001, 0.01090215294864004], {8, + 16} -> Complex[ + 0.17517709237484644`, -0.06154015846071672], {8, 17} -> + Complex[0.4133927336972727, 0.07976232385832366], {8, 18} -> + Complex[0.20540256085717942`, 0.21077891354451453`], {8, + 19} -> Complex[-0.03742196764438921, \ +-0.030128326443724755`], {8, 20} -> + Complex[-0.19605911325305342`, -0.18681308826202053`], {9, + 10} -> Complex[0.6738621924455607, 0.], {9, 11} -> + Complex[2.42861286636753*^-17, -7.979727989493313*^-17], { + 9, 12} -> Complex[0.6738621924455607, 0.], {9, 13} -> + Complex[-0.04268060967745001, -0.023059250426567146`], {9, + 14} -> Complex[-0.10752531345447021`, + 0.09271398750024093], {9, 15} -> + Complex[0.24953605886718944`, 0.23726268416179194`], {9, + 16} -> Complex[-0.24869282883169574`, + 0.1401584079854436], {9, 17} -> + Complex[0.00583211233850154, -0.10116953344808105`], {9, + 18} -> Complex[ + 0.23618815860033152`, 0.40921847295300184`], {9, 19} -> + Complex[0.5835728906428612, 0.5446721585854524], {9, 20} -> + Complex[0.05980282747160885, -0.07007653209262724], {10, + 11} -> Complex[0.3261378075544396, 0.], {10, 12} -> + Complex[2.42861286636753*^-17, 7.979727989493313*^-17], { + 10, 13} -> + Complex[0.10752531345447036`, 0.09271398750024114], {10, + 14} -> Complex[ + 0.04268060967745012, -0.023059250426567188`], {10, 15} -> + Complex[0.04667717945518325, 0.1531544506199518], {10, + 16} -> Complex[ + 0.11612240177174633`, -0.07247847487914186], {10, 17} -> + Complex[-0.15597490507210748`, 0.0891477723521908], {10, + 18} -> Complex[0.10349665614016897`, 0.197248245871046], { + 10, 19} -> + Complex[-0.01766294324618717, -0.07629791036757183], {10, 20} -> Complex[ - 0.05515660930576871, 0.00012490783968316282`], {14, 21} -> - Complex[-0.026634445818647365`, -0.271246719799264], {14, - 22} -> Complex[-0.0009452962128001206, \ --1.8498670906598045`*^-6], {14, 23} -> - Complex[-0.0013060825544481437`, 0.27807929010804266`], { - 14, 24} -> - Complex[6.938893903907228*^-18, -0.27807929010804266`], { - 15, 16} -> Complex[0.1547318858265081, 0.], {15, 17} -> - Complex[0.05139730785671253, -0.018926590336789572`], {15, - 18} -> - Complex[-0.05378013717194569, -0.0004103023339935032], { + 0.4136309863513148, -0.32459119381949325`], {11, 12} -> + Complex[0.6738621924455607, 0.], {11, 13} -> + Complex[-0.04268060967745001, -0.023059250426567146`], { + 11, 14} -> + Complex[-0.10752531345447021`, 0.09271398750024093], {11, + 15} -> Complex[ + 0.24953605886718944`, 0.23726268416179194`], {11, 16} -> + Complex[-0.24869282883169574`, 0.1401584079854436], {11, + 17} -> Complex[ + 0.00583211233850154, -0.10116953344808105`], {11, 18} -> + Complex[0.23618815860033152`, 0.40921847295300184`], {11, + 19} -> Complex[0.5835728906428612, 0.5446721585854524], { + 11, 20} -> + Complex[0.05980282747160885, -0.07007653209262724], {12, + 13} -> Complex[ + 0.10752531345447036`, 0.09271398750024114], {12, 14} -> + Complex[0.04268060967745012, -0.023059250426567188`], {12, + 15} -> Complex[ + 0.04667717945518325, 0.1531544506199518], {12, 16} -> + Complex[0.11612240177174633`, -0.07247847487914186], {12, + 17} -> Complex[-0.15597490507210748`, + 0.0891477723521908], {12, 18} -> + Complex[0.10349665614016897`, 0.197248245871046], {12, + 19} -> Complex[-0.01766294324618717, \ +-0.07629791036757183], {12, 20} -> + Complex[0.4136309863513148, -0.32459119381949325`], {13, + 14} -> Complex[0.4155199590831424, 0.], {13, 15} -> + Complex[0.10286685511517638`, -0.0766374954583717], {13, + 16} -> Complex[ + 0.15338653623116225`, -0.06791293700745762], {13, 17} -> + Complex[-0.3323619403479582, 0.050870880986855216`], {13, + 18} -> Complex[ + 0.06560512137155002, 0.16228715013884426`], {13, 19} -> + Complex[0.05020845049867415, -0.26771648586524205`], {13, + 20} -> Complex[-0.021712677276518863`, + 0.058041075609157985`], {14, 15} -> + Complex[-0.02984323129946001, 0.01090215294864004], {14, + 16} -> Complex[ + 0.17517709237484644`, -0.06154015846071672], {14, 17} -> + Complex[0.4133927336972727, 0.07976232385832366], {14, + 18} -> Complex[ + 0.20540256085717942`, 0.21077891354451453`], {14, 19} -> + Complex[-0.03742196764438921, -0.030128326443724755`], { + 14, 20} -> + Complex[-0.19605911325305342`, -0.18681308826202053`], { + 15, 16} -> + Complex[-0.30539523194520163`, 0.10550264339992513`], {15, + 17} -> + Complex[0.12212877058108958`, -0.09005118002308299], {15, + 18} -> Complex[0.04975850381773859, 0.0637738883370343], { 15, 19} -> - Complex[0.02663444581864737, 0.2712467197992639], {15, - 20} -> Complex[-0.0009452962128001213, \ --1.849867090669304*^-6], {15, 21} -> - Complex[-0.000017636699472665246`, \ --0.00012383773397450998`], {15, 22} -> - Complex[0.05515660930576866, 0.00012490783968316095`], { - 15, 23} -> - Complex[7.894016335854154*^-6, -0.008602132075960674], { - 15, 24} -> - Complex[-7.89401633587497*^-6, - 1.5178830414797064`*^-18], {16, 17} -> - Complex[0.05379673174522864, -0.0007416401460346826], {16, - 18} -> - Complex[-0.051397296777308234`, -0.01859492038010707], { - 16, 19} -> - Complex[-0.03773674551714698, 0.19081560591987778`], {16, - 20} -> Complex[-0.026634445818647372`, - 0.0778365212048926], {16, 21} -> - Complex[0.8470224364690501, 0.1883181155038509], {16, 22} -> - Complex[ - 0.000016933675279148797`, -0.00012383773397451302`], {16, - 23} -> Complex[ - 0.0012981885381123139`, -0.008523700770098444], {16, 24} -> - Complex[0.7859004070182107, 0.2609534572619835], {17, 18} -> - Complex[0.03895158675127773, 0.], {17, 19} -> - Complex[-0.019439452822070994`, -0.1924766868815898], {17, - 20} -> - Complex[-0.0002598580324258793, -0.003888044555498485], { - 17, 21} -> - Complex[-0.01885591810241921, 0.00450579754225058], {17, - 22} -> Complex[-0.019179268175876747`, - 0.0000390632327097088], {17, 23} -> - Complex[0.03920689940953653, 0.], {17, 24} -> - Complex[7.0302419351298*^-7, 0.002613753063531294], {18, - 19} -> Complex[-0.009673644515389308, - 0.09494936515397917], {18, 20} -> - Complex[0.01942284716938378, -0.09330760721542969], {18, - 21} -> Complex[ - 0.019195862749159683`, -0.0002922745793314728], {18, 22} -> - Complex[0.0188559291818235, 0.004174127585568078], {18, - 23} -> Complex[ - 0.9411238678566561, 0.19341019859437134`], {18, 24} -> - Complex[0.0002620773276979446, -0.002603237866672337], { - 19, 20} -> Complex[0.019669123004374556`, 0.], {19, 21} -> - Complex[-0.009768833833459008, -0.09670511030136167], {19, - 22} -> Complex[-0.00006578280908219912, - 7.28006620191595*^-19], {19, 23} -> - Complex[-0.000016605652687225828`, 0.09916907966616018], { - 19, 24} -> - Complex[1.6727401305043305`*^-18, -0.09916907966616016], { - 20, 21} -> - Complex[0.00006578280908219793, 4.254244153408405*^-17], { - 20, 22} -> - Complex[0.00976883383345901, -0.09670511030136163], {20, - 23} -> Complex[-0.009933502547815196, - 0.09883740970947764], {20, 24} -> - Complex[-0.009916907974532265, 0.09883807384038253], {21, - 22} -> Complex[0.01966912300437454, 0.], {21, 23} -> - Complex[ - 1.1079404294454619`*^-8, -0.0003316699566825008], {21, - 24} -> Complex[-1.107940429399415*^-8, - 2.241454781417565*^-19], {22, 23} -> - Complex[ - 0.000016594573282932862`, -0.0003313378120411798], {22, - 24} -> Complex[0.9751371712736823, 0.09850607189743646]}, - + Complex[0.26443368318488625`, 0.4306426521007979], {15, + 20} -> Complex[-0.14622966282928, 0.4816024306801036], { + 16, 17} -> + Complex[0.07419540882704287, 0.1062908220925275], {16, + 18} -> Complex[0.7513874079151734, 0.], {16, 19} -> + Complex[0.4498457818107638, -0.2438031767021491], {16, + 20} -> Complex[-0.11512981251395683`, \ +-0.25578858607648636`], {17, 18} -> + Complex[0.04363468584244643, 0.017866185612945815`], {17, + 19} -> Complex[-0.1520647355532879, + 0.12084858348643956`], {17, 20} -> + Complex[0.32171504751621155`, 0.15725504484569033`], {18, + 19} -> Complex[-0.04889639003422308, \ +-0.35866324581115683`], {18, 20} -> + Complex[0.08824078917592872, -0.0003106339104423192]}, Antisymmetric[{1, 2}]}]], Selectable -> False, Editable -> False, SelectWithContents -> True], "SummaryItem"]}]}}, GridBoxAlignment -> { @@ -11313,402 +6923,280 @@ RZ1D5dXfAH40arI= Q3`WickState[{ $CellContext`c[1, 3, 0], $CellContext`c[1, 1, 0], - $CellContext`c[2, 3, 1], - $CellContext`c[2, 3, 0], - $CellContext`c[2, 2, 1], $CellContext`c[2, 2, 0], - $CellContext`c[3, 1, 1], - $CellContext`c[3, 1, 0], - $CellContext`c[4, 3, 1], - $CellContext`c[4, 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Complex[-0.009673644515389308, -0.09494936515397917], {6, 8} -> - Complex[-0.019439452822070994`, 0.1924766868815898], {6, 9} -> - Complex[-0.03773674551714698, -0.19081560591987778`], {6, 10} -> - Complex[0.02663444581864737, -0.2712467197992639], {6, 11} -> - Complex[0.000017636699472639164`, -0.0001238377339745129], {6, 12} -> - Complex[0.8470224364690492, -0.18831811550385083`], {6, 13} -> - Complex[0.000017636699472639164`, -0.0001238377339745129], {6, 14} -> - Complex[0.8470224364690492, -0.18831811550385083`], {6, 15} -> - Complex[-0.03773674551714698, -0.19081560591987778`], {6, 16} -> - Complex[0.02663444581864737, -0.2712467197992639], {6, 17} -> - Complex[-0.009673644515389308, -0.09494936515397917], {6, 18} -> - Complex[-0.019439452822070994`, 0.1924766868815898], {6, 19} -> - Complex[0.9803308769956256, 0.], {6, 20} -> - Complex[-1.552346694521116*^-18, 1.2661074609920407`*^-19], {6, 21} -> - Complex[0.00006578280908219793, 4.254244153408405*^-17], {6, 22} -> - Complex[0.00976883383345901, -0.09670511030136163], {6, 23} -> - Complex[-0.009933502547815196, 0.09883740970947764], {6, 24} -> - Complex[-0.009916907974532265, 0.09883807384038253], {7, 8} -> - Complex[0.03895158675127773, 0.], {7, 9} -> - Complex[-0.051397296777308234`, 0.01859492038010707], {7, 10} -> - Complex[-0.05378013717194569, 0.0004103023339935032], {7, 11} -> - Complex[-0.0007781305712618784, -0.01090395795893278], {7, 12} -> - Complex[-0.05329448030995345, -0.16271619670523418`], {7, 13} -> - Complex[-0.0007781305712618784, -0.01090395795893278], {7, 14} -> - Complex[-0.05329448030995345, -0.16271619670523418`], {7, 15} -> - Complex[-0.051397296777308234`, 0.01859492038010707], {7, 16} -> - Complex[-0.05378013717194569, 0.0004103023339935032], {7, 17} -> - Complex[-4.787980228365168*^-18, -2.648859758904743*^-18], {7, 18} -> - Complex[0.03895158675127773, 0.], {7, 19} -> - Complex[-0.019439452822070994`, -0.1924766868815898], {7, 20} -> - Complex[-0.0002598580324258793, -0.003888044555498485], {7, 21} -> - Complex[-0.01885591810241921, 0.00450579754225058], {7, 22} -> - Complex[-0.019179268175876747`, 0.0000390632327097088], {7, 23} -> - Complex[0.03920689940953653, 0.], {7, 24} -> - Complex[7.0302419351298*^-7, 0.002613753063531294], {8, 9} -> - Complex[0.05379673174522864, 0.0007416401460346826], {8, 10} -> - Complex[0.05139730785671253, 0.018926590336789572`], {8, 11} -> - Complex[0.05327787465726625, -0.2618852763713944], {8, 12} -> - Complex[-0.00915537197655327, -0.10974136766841047`], {8, 13} -> - Complex[0.05327787465726625, -0.2618852763713944], {8, 14} -> - Complex[-0.00915537197655327, -0.10974136766841047`], {8, 15} -> - Complex[0.05379673174522864, 0.0007416401460346826], {8, 16} -> - Complex[0.05139730785671253, 0.018926590336789572`], {8, 17} -> - Complex[0.9610484132487226, 0.], {8, 18} -> - Complex[-4.787980228365168*^-18, 2.648859758904743*^-18], {8, 19} -> - Complex[-0.009673644515389308, 0.09494936515397917], {8, 20} -> - Complex[0.01942284716938378, -0.09330760721542969], {8, 21} -> - Complex[0.019195862749159683`, -0.0002922745793314728], {8, 22} -> - Complex[0.0188559291818235, 0.004174127585568078], {8, 23} -> - Complex[0.9411238678566561, 0.19341019859437134`], {8, 24} -> - Complex[0.0002620773276979446, -0.002603237866672337], {9, 10} -> - Complex[0.1547318858265081, 0.], {9, 11} -> - Complex[-0.004784170182671576, -2.079127072330406*^-17], {9, 12} -> - Complex[0.07261962399452813, 0.21892155188592954`], {9, 13} -> - Complex[-0.004784170182671576, -2.079127072330406*^-17], {9, 14} -> - Complex[0.07261962399452813, 0.21892155188592954`], {9, 15} -> - Complex[4.163336342344337*^-17, 6.938893903907228*^-18], {9, 16} -> - Complex[0.1547318858265081, 0.], {9, 17} -> - Complex[0.05139730785671253, -0.018926590336789572`], {9, 18} -> - Complex[-0.05378013717194569, -0.0004103023339935032], {9, 19} -> - Complex[0.02663444581864737, 0.2712467197992639], {9, 20} -> - Complex[-0.0009452962128001213, -1.849867090669304*^-6], {9, 21} -> - Complex[-0.000017636699472665246`, -0.00012383773397450998`], {9, 22} -> - Complex[0.05515660930576866, 0.00012490783968316095`], {9, 23} -> - Complex[7.894016335854154*^-6, -0.008602132075960674], {9, 24} -> - Complex[-7.89401633587497*^-6, 1.5178830414797064`*^-18], {10, 11} -> - Complex[-0.07261962399452816, 0.21892155188592985`], {10, 12} -> - Complex[0.004784170182671579, -2.7273105648872874`*^-16], {10, 13} -> - Complex[-0.07261962399452816, 0.21892155188592985`], {10, 14} -> - Complex[0.004784170182671579, -2.7273105648872874`*^-16], {10, 15} -> - Complex[0.8452681141734937, 0.], {10, 16} -> - Complex[4.163336342344337*^-17, -6.938893903907228*^-18], {10, 17} -> - Complex[0.05379673174522864, -0.0007416401460346826], {10, 18} -> - Complex[-0.051397296777308234`, -0.01859492038010707], {10, 19} -> - Complex[-0.03773674551714698, 0.19081560591987778`], {10, 20} -> - Complex[-0.026634445818647372`, 0.0778365212048926], {10, 21} -> - Complex[0.8470224364690501, 0.1883181155038509], {10, 22} -> - Complex[0.000016933675279148797`, -0.00012383773397451302`], {10, 23} -> - Complex[0.0012981885381123139`, -0.008523700770098444], {10, 24} -> - Complex[0.7859004070182107, 0.2609534572619835], {11, 12} -> - Complex[0.8452681141734925, 0.], {11, 13} -> - Complex[-2.901596064114331*^-17, -1.0211829212097848`*^-17], {11, 14} -> - Complex[0.8452681141734925, 0.], {11, 15} -> - Complex[0.004784170182671579, 2.7273105648872874`*^-16], {11, 16} -> - Complex[0.07261962399452813, -0.21892155188592954`], {11, 17} -> - Complex[-0.00915537197655327, 0.10974136766841047`], {11, 18} -> - Complex[-0.05329448030995345, 0.16271619670523418`], {11, 19} -> - Complex[0.8470224364690492, 0.18831811550385083`], {11, 20} -> - Complex[-0.00001693367527912708, 0.0001238377339745108], {11, 21} -> - Complex[-0.03773674551714702, 0.190815605919878], {11, 22} -> - Complex[0.026634445818647358`, -0.0778365212048925], {11, 23} -> - Complex[-0.08472202645669408, 0.26947715803208194`], {11, 24} -> - Complex[-0.08342383791858181, 0.26963368038096003`], {12, 13} -> - Complex[0.15473188582650818`, 0.], {12, 14} -> - Complex[-2.901596064114331*^-17, 1.0211829212097848`*^-17], {12, 15} -> - Complex[-0.07261962399452816, -0.21892155188592985`], {12, 16} -> - Complex[-0.004784170182671576, 2.079127072330406*^-17], {12, 17} -> - Complex[0.05327787465726625, 0.2618852763713944], {12, 18} -> - Complex[-0.0007781305712618784, 0.01090395795893278], {12, 19} -> - Complex[0.000017636699472639164`, 0.0001238377339745129], {12, 20} -> - Complex[0.05515660930576871, 0.00012490783968316282`], {12, 21} -> - Complex[-0.026634445818647365`, -0.271246719799264], {12, 22} -> - Complex[-0.0009452962128001206, -1.8498670906598045`*^-6], {12, 23} -> - Complex[-0.0013060825544481437`, 0.27807929010804266`], {12, 24} -> - Complex[6.938893903907228*^-18, -0.27807929010804266`], {13, 14} -> - Complex[0.8452681141734925, 0.], {13, 15} -> - Complex[0.004784170182671579, 2.7273105648872874`*^-16], {13, 16} -> - Complex[0.07261962399452813, -0.21892155188592954`], {13, 17} -> - Complex[-0.00915537197655327, 0.10974136766841047`], {13, 18} -> - Complex[-0.05329448030995345, 0.16271619670523418`], {13, 19} -> - Complex[0.8470224364690492, 0.18831811550385083`], {13, 20} -> - Complex[-0.00001693367527912708, 0.0001238377339745108], {13, 21} -> - Complex[-0.03773674551714702, 0.190815605919878], {13, 22} -> - Complex[0.026634445818647358`, -0.0778365212048925], {13, 23} -> - Complex[-0.08472202645669408, 0.26947715803208194`], {13, 24} -> - Complex[-0.08342383791858181, 0.26963368038096003`], {14, 15} -> - Complex[-0.07261962399452816, -0.21892155188592985`], {14, 16} -> - Complex[-0.004784170182671576, 2.079127072330406*^-17], {14, 17} -> - Complex[0.05327787465726625, 0.2618852763713944], {14, 18} -> - Complex[-0.0007781305712618784, 0.01090395795893278], {14, 19} -> - Complex[0.000017636699472639164`, 0.0001238377339745129], {14, 20} -> - Complex[0.05515660930576871, 0.00012490783968316282`], {14, 21} -> - Complex[-0.026634445818647365`, -0.271246719799264], {14, 22} -> - Complex[-0.0009452962128001206, -1.8498670906598045`*^-6], {14, 23} -> - Complex[-0.0013060825544481437`, 0.27807929010804266`], {14, 24} -> - Complex[6.938893903907228*^-18, -0.27807929010804266`], {15, 16} -> - Complex[0.1547318858265081, 0.], {15, 17} -> - Complex[0.05139730785671253, -0.018926590336789572`], {15, 18} -> - Complex[-0.05378013717194569, -0.0004103023339935032], {15, 19} -> - Complex[0.02663444581864737, 0.2712467197992639], {15, 20} -> - Complex[-0.0009452962128001213, -1.849867090669304*^-6], {15, 21} -> - Complex[-0.000017636699472665246`, -0.00012383773397450998`], {15, 22} -> - Complex[0.05515660930576866, 0.00012490783968316095`], {15, 23} -> - Complex[7.894016335854154*^-6, -0.008602132075960674], {15, 24} -> - Complex[-7.89401633587497*^-6, 1.5178830414797064`*^-18], {16, 17} -> - Complex[0.05379673174522864, -0.0007416401460346826], {16, 18} -> - Complex[-0.051397296777308234`, -0.01859492038010707], {16, 19} -> - Complex[-0.03773674551714698, 0.19081560591987778`], {16, 20} -> - Complex[-0.026634445818647372`, 0.0778365212048926], {16, 21} -> - Complex[0.8470224364690501, 0.1883181155038509], {16, 22} -> - Complex[0.000016933675279148797`, -0.00012383773397451302`], {16, 23} -> - Complex[0.0012981885381123139`, -0.008523700770098444], {16, 24} -> - Complex[0.7859004070182107, 0.2609534572619835], {17, 18} -> - Complex[0.03895158675127773, 0.], {17, 19} -> - Complex[-0.019439452822070994`, -0.1924766868815898], {17, 20} -> - Complex[-0.0002598580324258793, -0.003888044555498485], {17, 21} -> - Complex[-0.01885591810241921, 0.00450579754225058], {17, 22} -> - Complex[-0.019179268175876747`, 0.0000390632327097088], {17, 23} -> - Complex[0.03920689940953653, 0.], {17, 24} -> - Complex[7.0302419351298*^-7, 0.002613753063531294], {18, 19} -> - Complex[-0.009673644515389308, 0.09494936515397917], {18, 20} -> - Complex[0.01942284716938378, -0.09330760721542969], {18, 21} -> - Complex[0.019195862749159683`, -0.0002922745793314728], {18, 22} -> - Complex[0.0188559291818235, 0.004174127585568078], {18, 23} -> - Complex[0.9411238678566561, 0.19341019859437134`], {18, 24} -> - Complex[0.0002620773276979446, -0.002603237866672337], {19, 20} -> - Complex[0.019669123004374556`, 0.], {19, 21} -> - Complex[-0.009768833833459008, -0.09670511030136167], {19, 22} -> - Complex[-0.00006578280908219912, 7.28006620191595*^-19], {19, 23} -> - Complex[-0.000016605652687225828`, 0.09916907966616018], {19, 24} -> - Complex[1.6727401305043305`*^-18, -0.09916907966616016], {20, 21} -> - Complex[0.00006578280908219793, 4.254244153408405*^-17], {20, 22} -> - Complex[0.00976883383345901, -0.09670511030136163], {20, 23} -> - Complex[-0.009933502547815196, 0.09883740970947764], {20, 24} -> - Complex[-0.009916907974532265, 0.09883807384038253], {21, 22} -> - Complex[0.01966912300437454, 0.], {21, 23} -> - Complex[1.1079404294454619`*^-8, -0.0003316699566825008], {21, 24} -> - Complex[-1.107940429399415*^-8, 2.241454781417565*^-19], {22, 23} -> - Complex[0.000016594573282932862`, -0.0003313378120411798], {22, 24} -> - Complex[0.9751371712736823, 0.09850607189743646]}, + Complex[-0.04889639003422308, 0.35866324581115683`], {2, 4} -> + Complex[-0.1520647355532879, -0.12084858348643956`], {2, 5} -> + Complex[0.4498457818107638, 0.2438031767021491], {2, 6} -> + Complex[0.26443368318488625`, -0.4306426521007979], {2, 7} -> + Complex[-0.03742196764438921, 0.030128326443724755`], {2, 8} -> + Complex[0.05020845049867415, 0.26771648586524205`], {2, 9} -> + Complex[-0.01766294324618717, 0.07629791036757183], {2, 10} -> + Complex[0.5835728906428612, -0.5446721585854524], {2, 11} -> + Complex[-0.01766294324618717, 0.07629791036757183], {2, 12} -> + Complex[0.5835728906428612, -0.5446721585854524], {2, 13} -> + Complex[-0.03742196764438921, 0.030128326443724755`], {2, 14} -> + Complex[0.05020845049867415, 0.26771648586524205`], {2, 15} -> + Complex[0.3910495342752539, 0.025967555373879958`], {2, 16} -> + Complex[-0.04889639003422308, 0.35866324581115683`], {2, 17} -> + Complex[-0.11376740018597191`, -0.19180011846315012`], {2, 18} -> + Complex[0.4498457818107638, 0.2438031767021491], {2, 19} -> 1, {3, 4} -> + Complex[0.04363468584244643, -0.017866185612945815`], {3, 5} -> + Complex[0.7513874079151734, 0.], {3, 6} -> + Complex[0.04975850381773859, -0.0637738883370343], {3, 7} -> + Complex[0.20540256085717942`, -0.21077891354451453`], {3, 8} -> + Complex[0.06560512137155002, -0.16228715013884426`], {3, 9} -> + Complex[0.10349665614016897`, -0.197248245871046], {3, 10} -> + Complex[0.23618815860033152`, -0.40921847295300184`], {3, 11} -> + Complex[0.10349665614016897`, -0.197248245871046], {3, 12} -> + Complex[0.23618815860033152`, -0.40921847295300184`], {3, 13} -> + Complex[0.20540256085717942`, -0.21077891354451453`], {3, 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$CellContext`c[3]]]}], "}"}], "]"}]], "Output", + CellLabel->"Out[4]=", + CellID->120119998,ExpressionUUID->"259df09b-ce09-44b2-820f-b9564c29b24d"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"new", "=", - RowBox[{"op", "[", "ws", "]"}]}]], "Input", + RowBox[{"prj", "**", "ws"}]}]], "Input", CellProlog:>Needs["Q3`"], - CellLabel->"In[4]:=", + CellLabel->"In[5]:=", CellID->1688977281,ExpressionUUID->"1f771229-20ae-4236-8994-c2d4580f77ce"], Cell[BoxData[ @@ -11940,105 +7428,81 @@ Cell[BoxData[ InterpretationBox[ SubscriptBox["c", - TemplateBox[{",", "\",\"", "2", "3", "1"}, - "RowWithSeparators"]], - $CellContext`c[2, 3, 1]], ",", - InterpretationBox[ - SubscriptBox["c", - - TemplateBox[{",", "\",\"", "2", "3", "0"}, - "RowWithSeparators"]], - $CellContext`c[2, 3, 0]], ",", - InterpretationBox[ - SubscriptBox["c", - - TemplateBox[{",", "\",\"", "2", "2", "1"}, - "RowWithSeparators"]], - $CellContext`c[2, 2, 1]], ",", - InterpretationBox[ - SubscriptBox["c", - TemplateBox[{",", 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-> + Complex[0.17517709237484644`, 0.06154015846071672], {5, + 14} -> Complex[ + 0.15338653623116225`, 0.06791293700745762], {5, 15} -> + Complex[-0.04975850381773862, -0.06377388833703419], {5, + 16} -> Complex[0.24861259208482658`, 0.], {5, 17} -> + Complex[-0.043634685842446425`, -0.017866185612945742`], { + 5, 18} -> Complex[ + 2.0816681711721685`*^-17, 5.551115123125783*^-17], {5, + 19} -> Complex[-0.04889639003422308, \ +-0.35866324581115683`], {5, 20} -> + Complex[0.08824078917592872, -0.0003106339104423192], {6, + 7} -> Complex[-0.02984323129946001, \ +-0.01090215294864004], {6, 8} -> + Complex[0.10286685511517638`, 0.0766374954583717], {6, 9} -> + Complex[0.04667717945518325, -0.1531544506199518], {6, + 10} -> Complex[ + 0.24953605886718944`, -0.23726268416179194`], {6, 11} -> + Complex[0.04667717945518325, -0.1531544506199518], {6, 12} -> Complex[ - 0.004784170182671579, -2.7273105648872874`*^-16], {10, - 13} -> Complex[-0.07261962399452816, - 0.21892155188592985`], {10, 14} -> - Complex[ - 0.004784170182671579, -2.7273105648872874`*^-16], {10, - 15} -> Complex[0.8452681141734937, 0.], {10, 16} -> - Complex[ - 4.163336342344337*^-17, -6.938893903907228*^-18], {10, - 17} -> Complex[ - 0.05379673174522864, -0.0007416401460346826], {10, 18} -> - Complex[-0.051397296777308234`, -0.01859492038010707], { - 10, 19} -> - Complex[-0.03773674551714698, 0.19081560591987778`], {10, - 20} -> Complex[-0.026634445818647372`, - 0.0778365212048926], {10, 21} -> - Complex[0.8470224364690501, 0.1883181155038509], {10, 22} -> - Complex[ - 0.000016933675279148797`, -0.00012383773397451302`], {10, - 23} -> Complex[ - 0.0012981885381123139`, -0.008523700770098444], {10, 24} -> - Complex[0.7859004070182107, 0.2609534572619835], {11, 12} -> - Complex[0.8452681141734925, 0.], {11, 13} -> - Complex[-2.901596064114331*^-17, \ --1.0211829212097848`*^-17], {11, 14} -> - Complex[0.8452681141734925, 0.], {11, 15} -> - Complex[0.004784170182671579, 2.7273105648872874`*^-16], { - 11, 16} -> - Complex[0.07261962399452813, -0.21892155188592954`], {11, - 17} -> Complex[-0.00915537197655327, - 0.10974136766841047`], {11, 18} -> - Complex[-0.05329448030995345, 0.16271619670523418`], {11, - 19} -> Complex[ - 0.8470224364690492, 0.18831811550385083`], {11, 20} -> - Complex[-0.00001693367527912708, 0.0001238377339745108], { - 11, 21} -> - Complex[-0.03773674551714702, 0.190815605919878], {11, - 22} -> Complex[ - 0.026634445818647358`, -0.0778365212048925], {11, 23} -> - Complex[-0.08472202645669408, 0.26947715803208194`], {11, - 24} -> Complex[-0.08342383791858181, - 0.26963368038096003`], {12, 13} -> - Complex[0.15473188582650818`, 0.], {12, 14} -> - Complex[-2.901596064114331*^-17, - 1.0211829212097848`*^-17], {12, 15} -> - Complex[-0.07261962399452816, -0.21892155188592985`], {12, - 16} -> Complex[-0.004784170182671576, - 2.079127072330406*^-17], {12, 17} -> - Complex[0.05327787465726625, 0.2618852763713944], {12, - 18} -> Complex[-0.0007781305712618784, - 0.01090395795893278], {12, 19} -> - Complex[ - 0.000017636699472639164`, 0.0001238377339745129], {12, - 20} -> Complex[ - 0.05515660930576871, 0.00012490783968316282`], {12, 21} -> - Complex[-0.026634445818647365`, -0.271246719799264], {12, - 22} -> Complex[-0.0009452962128001206, \ --1.8498670906598045`*^-6], {12, 23} -> - Complex[-0.0013060825544481437`, 0.27807929010804266`], { - 12, 24} -> - Complex[6.938893903907228*^-18, -0.27807929010804266`], { - 13, 14} -> Complex[0.8452681141734925, 0.], {13, 15} -> - Complex[0.004784170182671579, 2.7273105648872874`*^-16], { - 13, 16} -> - Complex[0.07261962399452813, -0.21892155188592954`], {13, - 17} -> Complex[-0.00915537197655327, - 0.10974136766841047`], {13, 18} -> - Complex[-0.05329448030995345, 0.16271619670523418`], {13, + 0.24953605886718944`, -0.23726268416179194`], {6, 13} -> + Complex[-0.02984323129946001, -0.01090215294864004], {6, + 14} -> Complex[ + 0.10286685511517638`, 0.0766374954583717], {6, 15} -> + Complex[0.3009141283362328, 0.], {6, 16} -> + Complex[-0.04975850381773862, 0.06377388833703419], {6, + 17} -> Complex[-0.15900572219376757`, + 0.05750044204194605], {6, 18} -> + Complex[0.30539523194520163`, 0.10550264339992507`], {6, 19} -> Complex[ - 0.8470224364690492, 0.18831811550385083`], {13, 20} -> - Complex[-0.00001693367527912708, 0.0001238377339745108], { - 13, 21} -> - Complex[-0.03773674551714702, 0.190815605919878], {13, - 22} -> Complex[ - 0.026634445818647358`, -0.0778365212048925], {13, 23} -> - Complex[-0.08472202645669408, 0.26947715803208194`], {13, - 24} -> Complex[-0.08342383791858181, - 0.26963368038096003`], {14, 15} -> - Complex[-0.07261962399452816, -0.21892155188592985`], {14, - 16} -> Complex[-0.004784170182671576, - 2.079127072330406*^-17], {14, 17} -> - Complex[0.05327787465726625, 0.2618852763713944], {14, - 18} -> Complex[-0.0007781305712618784, - 0.01090395795893278], {14, 19} -> - Complex[ - 0.000017636699472639164`, 0.0001238377339745129], {14, + 0.3910495342752539, -0.025967555373879958`], {6, 20} -> + Complex[-0.0981351838004039, -0.2911740023151237], {7, 8} -> + Complex[0.4155199590831424, 0.], {7, 9} -> + Complex[0.04268060967745012, 0.023059250426567188`], {7, + 10} -> Complex[-0.10752531345447021`, \ +-0.09271398750024093], {7, 11} -> + Complex[0.04268060967745012, 0.023059250426567188`], {7, + 12} -> Complex[-0.10752531345447021`, \ +-0.09271398750024093], {7, 13} -> + Complex[-6.938893903907228*^-18, \ +-2.7755575615628914`*^-17], {7, 14} -> + Complex[0.4155199590831424, 0.], {7, 15} -> + Complex[0.10286685511517638`, -0.0766374954583717], {7, + 16} -> Complex[ + 0.15338653623116225`, -0.06791293700745762], {7, 17} -> + Complex[-0.3323619403479582, 0.050870880986855216`], {7, + 18} -> Complex[ + 0.06560512137155002, 0.16228715013884426`], {7, 19} -> + Complex[0.05020845049867415, -0.26771648586524205`], {7, + 20} -> Complex[-0.021712677276518863`, + 0.058041075609157985`], {8, 9} -> + Complex[0.10752531345447036`, -0.09271398750024114], {8, + 10} -> Complex[-0.04268060967745001, + 0.023059250426567146`], {8, 11} -> + Complex[0.10752531345447036`, -0.09271398750024114], {8, + 12} -> Complex[-0.04268060967745001, + 0.023059250426567146`], {8, 13} -> + Complex[0.5844800409168576, 0.], {8, 14} -> + Complex[-6.938893903907228*^-18, + 2.7755575615628914`*^-17], {8, 15} -> + Complex[-0.02984323129946001, 0.01090215294864004], {8, + 16} -> Complex[ + 0.17517709237484644`, -0.06154015846071672], {8, 17} -> + Complex[0.4133927336972727, 0.07976232385832366], {8, 18} -> + Complex[0.20540256085717942`, 0.21077891354451453`], {8, + 19} -> Complex[-0.03742196764438921, \ +-0.030128326443724755`], {8, 20} -> + Complex[-0.19605911325305342`, -0.18681308826202053`], {9, + 10} -> Complex[0.6738621924455607, 0.], {9, 11} -> + Complex[2.42861286636753*^-17, -7.979727989493313*^-17], { + 9, 12} -> Complex[0.6738621924455607, 0.], {9, 13} -> + Complex[-0.04268060967745001, -0.023059250426567146`], {9, + 14} -> Complex[-0.10752531345447021`, + 0.09271398750024093], {9, 15} -> + Complex[0.24953605886718944`, 0.23726268416179194`], {9, + 16} -> Complex[-0.24869282883169574`, + 0.1401584079854436], {9, 17} -> + Complex[0.00583211233850154, -0.10116953344808105`], {9, + 18} -> Complex[ + 0.23618815860033152`, 0.40921847295300184`], {9, 19} -> + Complex[0.5835728906428612, 0.5446721585854524], {9, 20} -> + Complex[0.05980282747160885, -0.07007653209262724], {10, + 11} -> Complex[0.3261378075544396, 0.], {10, 12} -> + Complex[2.42861286636753*^-17, 7.979727989493313*^-17], { + 10, 13} -> + Complex[0.10752531345447036`, 0.09271398750024114], {10, + 14} -> Complex[ + 0.04268060967745012, -0.023059250426567188`], {10, 15} -> + Complex[0.04667717945518325, 0.1531544506199518], {10, + 16} -> Complex[ + 0.11612240177174633`, -0.07247847487914186], {10, 17} -> + Complex[-0.15597490507210748`, 0.0891477723521908], {10, + 18} -> Complex[0.10349665614016897`, 0.197248245871046], { + 10, 19} -> + Complex[-0.01766294324618717, -0.07629791036757183], {10, 20} -> Complex[ - 0.05515660930576871, 0.00012490783968316282`], {14, 21} -> - Complex[-0.026634445818647365`, -0.271246719799264], {14, - 22} -> Complex[-0.0009452962128001206, \ --1.8498670906598045`*^-6], {14, 23} -> - Complex[-0.0013060825544481437`, 0.27807929010804266`], { - 14, 24} -> - Complex[6.938893903907228*^-18, -0.27807929010804266`], { - 15, 16} -> Complex[0.1547318858265081, 0.], {15, 17} -> - Complex[0.05139730785671253, -0.018926590336789572`], {15, - 18} -> - Complex[-0.05378013717194569, -0.0004103023339935032], { + 0.4136309863513148, -0.32459119381949325`], {11, 12} -> + Complex[0.6738621924455607, 0.], {11, 13} -> + Complex[-0.04268060967745001, -0.023059250426567146`], { + 11, 14} -> + Complex[-0.10752531345447021`, 0.09271398750024093], {11, + 15} -> Complex[ + 0.24953605886718944`, 0.23726268416179194`], {11, 16} -> + Complex[-0.24869282883169574`, 0.1401584079854436], {11, + 17} -> Complex[ + 0.00583211233850154, -0.10116953344808105`], {11, 18} -> + Complex[0.23618815860033152`, 0.40921847295300184`], {11, + 19} -> Complex[0.5835728906428612, 0.5446721585854524], { + 11, 20} -> + Complex[0.05980282747160885, -0.07007653209262724], {12, + 13} -> Complex[ + 0.10752531345447036`, 0.09271398750024114], {12, 14} -> + Complex[0.04268060967745012, -0.023059250426567188`], {12, + 15} -> Complex[ + 0.04667717945518325, 0.1531544506199518], {12, 16} -> + Complex[0.11612240177174633`, -0.07247847487914186], {12, + 17} -> Complex[-0.15597490507210748`, + 0.0891477723521908], {12, 18} -> + Complex[0.10349665614016897`, 0.197248245871046], {12, + 19} -> Complex[-0.01766294324618717, \ +-0.07629791036757183], {12, 20} -> + Complex[0.4136309863513148, -0.32459119381949325`], {13, + 14} -> Complex[0.4155199590831424, 0.], {13, 15} -> + Complex[0.10286685511517638`, -0.0766374954583717], {13, + 16} -> Complex[ + 0.15338653623116225`, -0.06791293700745762], {13, 17} -> + Complex[-0.3323619403479582, 0.050870880986855216`], {13, + 18} -> Complex[ + 0.06560512137155002, 0.16228715013884426`], {13, 19} -> + Complex[0.05020845049867415, -0.26771648586524205`], {13, + 20} -> Complex[-0.021712677276518863`, + 0.058041075609157985`], {14, 15} -> + Complex[-0.02984323129946001, 0.01090215294864004], {14, + 16} -> Complex[ + 0.17517709237484644`, -0.06154015846071672], {14, 17} -> + Complex[0.4133927336972727, 0.07976232385832366], {14, + 18} -> Complex[ + 0.20540256085717942`, 0.21077891354451453`], {14, 19} -> + Complex[-0.03742196764438921, -0.030128326443724755`], { + 14, 20} -> + Complex[-0.19605911325305342`, -0.18681308826202053`], { + 15, 16} -> + Complex[-0.30539523194520163`, 0.10550264339992513`], {15, + 17} -> + Complex[0.12212877058108958`, -0.09005118002308299], {15, + 18} -> Complex[0.04975850381773859, 0.0637738883370343], { 15, 19} -> - Complex[0.02663444581864737, 0.2712467197992639], {15, - 20} -> Complex[-0.0009452962128001213, \ --1.849867090669304*^-6], {15, 21} -> - Complex[-0.000017636699472665246`, \ --0.00012383773397450998`], {15, 22} -> - Complex[0.05515660930576866, 0.00012490783968316095`], { - 15, 23} -> - Complex[7.894016335854154*^-6, -0.008602132075960674], { - 15, 24} -> - Complex[-7.89401633587497*^-6, - 1.5178830414797064`*^-18], {16, 17} -> - Complex[0.05379673174522864, -0.0007416401460346826], {16, - 18} -> - Complex[-0.051397296777308234`, -0.01859492038010707], { - 16, 19} -> - Complex[-0.03773674551714698, 0.19081560591987778`], {16, - 20} -> Complex[-0.026634445818647372`, - 0.0778365212048926], {16, 21} -> - Complex[0.8470224364690501, 0.1883181155038509], {16, 22} -> - Complex[ - 0.000016933675279148797`, -0.00012383773397451302`], {16, - 23} -> Complex[ - 0.0012981885381123139`, -0.008523700770098444], {16, 24} -> - Complex[0.7859004070182107, 0.2609534572619835], {17, 18} -> - Complex[0.03895158675127773, 0.], {17, 19} -> - Complex[-0.019439452822070994`, -0.1924766868815898], {17, - 20} -> - Complex[-0.0002598580324258793, -0.003888044555498485], { - 17, 21} -> - Complex[-0.01885591810241921, 0.00450579754225058], {17, - 22} -> Complex[-0.019179268175876747`, - 0.0000390632327097088], {17, 23} -> - Complex[0.03920689940953653, 0.], {17, 24} -> - Complex[7.0302419351298*^-7, 0.002613753063531294], {18, - 19} -> Complex[-0.009673644515389308, - 0.09494936515397917], {18, 20} -> - Complex[0.01942284716938378, -0.09330760721542969], {18, - 21} -> Complex[ - 0.019195862749159683`, -0.0002922745793314728], {18, 22} -> - Complex[0.0188559291818235, 0.004174127585568078], {18, - 23} -> Complex[ - 0.9411238678566561, 0.19341019859437134`], {18, 24} -> - Complex[0.0002620773276979446, -0.002603237866672337], { - 19, 20} -> Complex[0.019669123004374556`, 0.], {19, 21} -> - Complex[-0.009768833833459008, -0.09670511030136167], {19, - 22} -> Complex[-0.00006578280908219912, - 7.28006620191595*^-19], {19, 23} -> - Complex[-0.000016605652687225828`, 0.09916907966616018], { - 19, 24} -> - Complex[1.6727401305043305`*^-18, -0.09916907966616016], { - 20, 21} -> - Complex[0.00006578280908219793, 4.254244153408405*^-17], { - 20, 22} -> - Complex[0.00976883383345901, -0.09670511030136163], {20, - 23} -> Complex[-0.009933502547815196, - 0.09883740970947764], {20, 24} -> - Complex[-0.009916907974532265, 0.09883807384038253], {21, - 22} -> Complex[0.01966912300437454, 0.], {21, 23} -> - Complex[ - 1.1079404294454619`*^-8, -0.0003316699566825008], {21, - 24} -> Complex[-1.107940429399415*^-8, - 2.241454781417565*^-19], {22, 23} -> - Complex[ - 0.000016594573282932862`, -0.0003313378120411798], {22, - 24} -> Complex[0.9751371712736823, 0.09850607189743646]}, - + Complex[0.26443368318488625`, 0.4306426521007979], {15, + 20} -> Complex[-0.14622966282928, 0.4816024306801036], { + 16, 17} -> + Complex[0.07419540882704287, 0.1062908220925275], {16, + 18} -> Complex[0.7513874079151734, 0.], {16, 19} -> + Complex[0.4498457818107638, -0.2438031767021491], {16, + 20} -> Complex[-0.11512981251395683`, \ +-0.25578858607648636`], {17, 18} -> + Complex[0.04363468584244643, 0.017866185612945815`], {17, + 19} -> Complex[-0.1520647355532879, + 0.12084858348643956`], {17, 20} -> + Complex[0.32171504751621155`, 0.15725504484569033`], {18, + 19} -> Complex[-0.04889639003422308, \ +-0.35866324581115683`], {18, 20} -> + Complex[0.08824078917592872, -0.0003106339104423192]}, Antisymmetric[{1, 2}]}]], Selectable -> False, Editable -> False, SelectWithContents -> True], "SummaryItem"]}]}}, GridBoxAlignment -> { @@ -14446,402 +9351,280 @@ RZ1D5dXfAH40arI= Q3`WickState[{ $CellContext`c[1, 3, 0], $CellContext`c[1, 1, 0], - $CellContext`c[2, 3, 1], - $CellContext`c[2, 3, 0], - $CellContext`c[2, 2, 1], $CellContext`c[2, 2, 0], - $CellContext`c[3, 1, 1], - $CellContext`c[3, 1, 0], - $CellContext`c[4, 3, 1], - $CellContext`c[4, 3, 0], - $CellContext`c[4, 2, 0], - $CellContext`c[4, 2, 1], - $CellContext`c[4, 2, 0], - $CellContext`c[4, 2, 1], - $CellContext`c[4, 3, 1], - 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0.00012383773397450998`], {4, 17} -> - Complex[0.019195862749159683`, 0.0002922745793314728], {4, 18} -> - Complex[-0.01885591810241921, -0.00450579754225058], {4, 19} -> - Complex[0.00006578280908219793, -4.254244153408405*^-17], {4, 20} -> - Complex[-0.009768833833459008, 0.09670511030136167], {4, 21} -> - Complex[0.980330876995626, 0.], {4, 22} -> - Complex[6.938893903907228*^-18, 4.336808689942018*^-19], {4, 23} -> - Complex[0.000016594573282932862`, -0.0003313378120411798], {4, 24} -> - Complex[0.9751371712736823, 0.09850607189743646], {5, 6} -> - Complex[0.019669123004374556`, 0.], {5, 7} -> - Complex[0.01942284716938378, 0.09330760721542969], {5, 8} -> - Complex[-0.0002598580324258793, 0.003888044555498485], {5, 9} -> - Complex[-0.026634445818647372`, -0.0778365212048926], {5, 10} -> - Complex[-0.0009452962128001213, 1.849867090669304*^-6], {5, 11} -> - Complex[0.05515660930576871, -0.00012490783968316282`], {5, 12} -> - Complex[-0.00001693367527912708, -0.0001238377339745108], {5, 13} -> - Complex[0.05515660930576871, -0.00012490783968316282`], {5, 14} -> - Complex[-0.00001693367527912708, -0.0001238377339745108], {5, 15} -> - Complex[-0.026634445818647372`, -0.0778365212048926], {5, 16} -> - Complex[-0.0009452962128001213, 1.849867090669304*^-6], {5, 17} -> - Complex[0.01942284716938378, 0.09330760721542969], {5, 18} -> - Complex[-0.0002598580324258793, 0.003888044555498485], {5, 19} -> - Complex[-1.552346694521116*^-18, -1.2661074609920407`*^-19], {5, 20} -> - Complex[0.019669123004374556`, 0.], {5, 21} -> - Complex[-0.009768833833459008, -0.09670511030136167], {5, 22} -> - Complex[-0.00006578280908219912, 7.28006620191595*^-19], {5, 23} -> - Complex[-0.000016605652687225828`, 0.09916907966616018], {5, 24} -> - Complex[1.6727401305043305`*^-18, -0.09916907966616016], {6, 7} -> - Complex[-0.009673644515389308, -0.09494936515397917], {6, 8} -> - Complex[-0.019439452822070994`, 0.1924766868815898], {6, 9} -> - Complex[-0.03773674551714698, -0.19081560591987778`], {6, 10} -> - Complex[0.02663444581864737, -0.2712467197992639], {6, 11} -> - Complex[0.000017636699472639164`, -0.0001238377339745129], {6, 12} -> - Complex[0.8470224364690492, -0.18831811550385083`], {6, 13} -> - Complex[0.000017636699472639164`, -0.0001238377339745129], {6, 14} -> - Complex[0.8470224364690492, -0.18831811550385083`], {6, 15} -> - Complex[-0.03773674551714698, -0.19081560591987778`], {6, 16} -> - Complex[0.02663444581864737, -0.2712467197992639], {6, 17} -> - Complex[-0.009673644515389308, -0.09494936515397917], {6, 18} -> - Complex[-0.019439452822070994`, 0.1924766868815898], {6, 19} -> - Complex[0.9803308769956256, 0.], {6, 20} -> - Complex[-1.552346694521116*^-18, 1.2661074609920407`*^-19], {6, 21} -> - Complex[0.00006578280908219793, 4.254244153408405*^-17], {6, 22} -> - Complex[0.00976883383345901, -0.09670511030136163], {6, 23} -> - Complex[-0.009933502547815196, 0.09883740970947764], {6, 24} -> - Complex[-0.009916907974532265, 0.09883807384038253], {7, 8} -> - Complex[0.03895158675127773, 0.], {7, 9} -> - Complex[-0.051397296777308234`, 0.01859492038010707], {7, 10} -> - Complex[-0.05378013717194569, 0.0004103023339935032], {7, 11} -> - Complex[-0.0007781305712618784, -0.01090395795893278], {7, 12} -> - Complex[-0.05329448030995345, -0.16271619670523418`], {7, 13} -> - Complex[-0.0007781305712618784, -0.01090395795893278], {7, 14} -> - Complex[-0.05329448030995345, -0.16271619670523418`], {7, 15} -> - Complex[-0.051397296777308234`, 0.01859492038010707], {7, 16} -> - Complex[-0.05378013717194569, 0.0004103023339935032], {7, 17} -> - Complex[-4.787980228365168*^-18, -2.648859758904743*^-18], {7, 18} -> - Complex[0.03895158675127773, 0.], {7, 19} -> - Complex[-0.019439452822070994`, -0.1924766868815898], {7, 20} -> - Complex[-0.0002598580324258793, -0.003888044555498485], {7, 21} -> - Complex[-0.01885591810241921, 0.00450579754225058], {7, 22} -> - Complex[-0.019179268175876747`, 0.0000390632327097088], {7, 23} -> - Complex[0.03920689940953653, 0.], {7, 24} -> - Complex[7.0302419351298*^-7, 0.002613753063531294], {8, 9} -> - Complex[0.05379673174522864, 0.0007416401460346826], {8, 10} -> - Complex[0.05139730785671253, 0.018926590336789572`], {8, 11} -> - Complex[0.05327787465726625, -0.2618852763713944], {8, 12} -> - Complex[-0.00915537197655327, -0.10974136766841047`], {8, 13} -> - Complex[0.05327787465726625, -0.2618852763713944], {8, 14} -> - Complex[-0.00915537197655327, -0.10974136766841047`], {8, 15} -> - Complex[0.05379673174522864, 0.0007416401460346826], {8, 16} -> - Complex[0.05139730785671253, 0.018926590336789572`], {8, 17} -> - Complex[0.9610484132487226, 0.], {8, 18} -> - Complex[-4.787980228365168*^-18, 2.648859758904743*^-18], {8, 19} -> - Complex[-0.009673644515389308, 0.09494936515397917], {8, 20} -> - Complex[0.01942284716938378, -0.09330760721542969], {8, 21} -> - Complex[0.019195862749159683`, -0.0002922745793314728], {8, 22} -> - Complex[0.0188559291818235, 0.004174127585568078], {8, 23} -> - Complex[0.9411238678566561, 0.19341019859437134`], {8, 24} -> - Complex[0.0002620773276979446, -0.002603237866672337], {9, 10} -> - Complex[0.1547318858265081, 0.], {9, 11} -> - Complex[-0.004784170182671576, -2.079127072330406*^-17], {9, 12} -> - Complex[0.07261962399452813, 0.21892155188592954`], {9, 13} -> - Complex[-0.004784170182671576, -2.079127072330406*^-17], {9, 14} -> - Complex[0.07261962399452813, 0.21892155188592954`], {9, 15} -> - Complex[4.163336342344337*^-17, 6.938893903907228*^-18], {9, 16} -> - Complex[0.1547318858265081, 0.], {9, 17} -> - Complex[0.05139730785671253, -0.018926590336789572`], {9, 18} -> - Complex[-0.05378013717194569, -0.0004103023339935032], {9, 19} -> - Complex[0.02663444581864737, 0.2712467197992639], {9, 20} -> - Complex[-0.0009452962128001213, -1.849867090669304*^-6], {9, 21} -> - Complex[-0.000017636699472665246`, -0.00012383773397450998`], {9, 22} -> - Complex[0.05515660930576866, 0.00012490783968316095`], {9, 23} -> - Complex[7.894016335854154*^-6, -0.008602132075960674], {9, 24} -> - Complex[-7.89401633587497*^-6, 1.5178830414797064`*^-18], {10, 11} -> - Complex[-0.07261962399452816, 0.21892155188592985`], {10, 12} -> - Complex[0.004784170182671579, -2.7273105648872874`*^-16], {10, 13} -> - Complex[-0.07261962399452816, 0.21892155188592985`], {10, 14} -> - Complex[0.004784170182671579, -2.7273105648872874`*^-16], {10, 15} -> - Complex[0.8452681141734937, 0.], {10, 16} -> - Complex[4.163336342344337*^-17, -6.938893903907228*^-18], {10, 17} -> - Complex[0.05379673174522864, -0.0007416401460346826], {10, 18} -> - Complex[-0.051397296777308234`, -0.01859492038010707], {10, 19} -> - Complex[-0.03773674551714698, 0.19081560591987778`], {10, 20} -> - Complex[-0.026634445818647372`, 0.0778365212048926], {10, 21} -> - Complex[0.8470224364690501, 0.1883181155038509], {10, 22} -> - Complex[0.000016933675279148797`, -0.00012383773397451302`], {10, 23} -> - Complex[0.0012981885381123139`, -0.008523700770098444], {10, 24} -> - Complex[0.7859004070182107, 0.2609534572619835], {11, 12} -> - Complex[0.8452681141734925, 0.], {11, 13} -> - Complex[-2.901596064114331*^-17, -1.0211829212097848`*^-17], {11, 14} -> - Complex[0.8452681141734925, 0.], {11, 15} -> - Complex[0.004784170182671579, 2.7273105648872874`*^-16], {11, 16} -> - Complex[0.07261962399452813, -0.21892155188592954`], {11, 17} -> - Complex[-0.00915537197655327, 0.10974136766841047`], {11, 18} -> - Complex[-0.05329448030995345, 0.16271619670523418`], {11, 19} -> - Complex[0.8470224364690492, 0.18831811550385083`], {11, 20} -> - Complex[-0.00001693367527912708, 0.0001238377339745108], {11, 21} -> - Complex[-0.03773674551714702, 0.190815605919878], {11, 22} -> - Complex[0.026634445818647358`, -0.0778365212048925], {11, 23} -> - Complex[-0.08472202645669408, 0.26947715803208194`], {11, 24} -> - Complex[-0.08342383791858181, 0.26963368038096003`], {12, 13} -> - Complex[0.15473188582650818`, 0.], {12, 14} -> - Complex[-2.901596064114331*^-17, 1.0211829212097848`*^-17], {12, 15} -> - Complex[-0.07261962399452816, -0.21892155188592985`], {12, 16} -> - Complex[-0.004784170182671576, 2.079127072330406*^-17], {12, 17} -> - Complex[0.05327787465726625, 0.2618852763713944], {12, 18} -> - Complex[-0.0007781305712618784, 0.01090395795893278], {12, 19} -> - Complex[0.000017636699472639164`, 0.0001238377339745129], {12, 20} -> - Complex[0.05515660930576871, 0.00012490783968316282`], {12, 21} -> - Complex[-0.026634445818647365`, -0.271246719799264], {12, 22} -> - Complex[-0.0009452962128001206, -1.8498670906598045`*^-6], {12, 23} -> - Complex[-0.0013060825544481437`, 0.27807929010804266`], {12, 24} -> - Complex[6.938893903907228*^-18, -0.27807929010804266`], {13, 14} -> - Complex[0.8452681141734925, 0.], {13, 15} -> - Complex[0.004784170182671579, 2.7273105648872874`*^-16], {13, 16} -> - Complex[0.07261962399452813, -0.21892155188592954`], {13, 17} -> - Complex[-0.00915537197655327, 0.10974136766841047`], {13, 18} -> - Complex[-0.05329448030995345, 0.16271619670523418`], {13, 19} -> - Complex[0.8470224364690492, 0.18831811550385083`], {13, 20} -> - Complex[-0.00001693367527912708, 0.0001238377339745108], {13, 21} -> - Complex[-0.03773674551714702, 0.190815605919878], {13, 22} -> - Complex[0.026634445818647358`, -0.0778365212048925], {13, 23} -> - Complex[-0.08472202645669408, 0.26947715803208194`], {13, 24} -> - Complex[-0.08342383791858181, 0.26963368038096003`], {14, 15} -> - Complex[-0.07261962399452816, -0.21892155188592985`], {14, 16} -> - Complex[-0.004784170182671576, 2.079127072330406*^-17], {14, 17} -> - Complex[0.05327787465726625, 0.2618852763713944], {14, 18} -> - Complex[-0.0007781305712618784, 0.01090395795893278], {14, 19} -> - Complex[0.000017636699472639164`, 0.0001238377339745129], {14, 20} -> - Complex[0.05515660930576871, 0.00012490783968316282`], {14, 21} -> - Complex[-0.026634445818647365`, -0.271246719799264], {14, 22} -> - Complex[-0.0009452962128001206, -1.8498670906598045`*^-6], {14, 23} -> - Complex[-0.0013060825544481437`, 0.27807929010804266`], {14, 24} -> - Complex[6.938893903907228*^-18, -0.27807929010804266`], {15, 16} -> - Complex[0.1547318858265081, 0.], {15, 17} -> - Complex[0.05139730785671253, -0.018926590336789572`], {15, 18} -> - Complex[-0.05378013717194569, -0.0004103023339935032], {15, 19} -> - Complex[0.02663444581864737, 0.2712467197992639], {15, 20} -> - Complex[-0.0009452962128001213, -1.849867090669304*^-6], {15, 21} -> - Complex[-0.000017636699472665246`, -0.00012383773397450998`], {15, 22} -> - Complex[0.05515660930576866, 0.00012490783968316095`], {15, 23} -> - Complex[7.894016335854154*^-6, -0.008602132075960674], {15, 24} -> - Complex[-7.89401633587497*^-6, 1.5178830414797064`*^-18], {16, 17} -> - Complex[0.05379673174522864, -0.0007416401460346826], {16, 18} -> - Complex[-0.051397296777308234`, -0.01859492038010707], {16, 19} -> - Complex[-0.03773674551714698, 0.19081560591987778`], {16, 20} -> - Complex[-0.026634445818647372`, 0.0778365212048926], {16, 21} -> - Complex[0.8470224364690501, 0.1883181155038509], {16, 22} -> - Complex[0.000016933675279148797`, -0.00012383773397451302`], {16, 23} -> - Complex[0.0012981885381123139`, -0.008523700770098444], {16, 24} -> - Complex[0.7859004070182107, 0.2609534572619835], {17, 18} -> - Complex[0.03895158675127773, 0.], {17, 19} -> - Complex[-0.019439452822070994`, -0.1924766868815898], {17, 20} -> - Complex[-0.0002598580324258793, -0.003888044555498485], {17, 21} -> - Complex[-0.01885591810241921, 0.00450579754225058], {17, 22} -> - Complex[-0.019179268175876747`, 0.0000390632327097088], {17, 23} -> - Complex[0.03920689940953653, 0.], {17, 24} -> - Complex[7.0302419351298*^-7, 0.002613753063531294], {18, 19} -> - Complex[-0.009673644515389308, 0.09494936515397917], {18, 20} -> - Complex[0.01942284716938378, -0.09330760721542969], {18, 21} -> - Complex[0.019195862749159683`, -0.0002922745793314728], {18, 22} -> - Complex[0.0188559291818235, 0.004174127585568078], {18, 23} -> - Complex[0.9411238678566561, 0.19341019859437134`], {18, 24} -> - Complex[0.0002620773276979446, -0.002603237866672337], {19, 20} -> - Complex[0.019669123004374556`, 0.], {19, 21} -> - Complex[-0.009768833833459008, -0.09670511030136167], {19, 22} -> - Complex[-0.00006578280908219912, 7.28006620191595*^-19], {19, 23} -> - Complex[-0.000016605652687225828`, 0.09916907966616018], {19, 24} -> - Complex[1.6727401305043305`*^-18, -0.09916907966616016], {20, 21} -> - Complex[0.00006578280908219793, 4.254244153408405*^-17], {20, 22} -> - Complex[0.00976883383345901, -0.09670511030136163], {20, 23} -> - Complex[-0.009933502547815196, 0.09883740970947764], {20, 24} -> - Complex[-0.009916907974532265, 0.09883807384038253], {21, 22} -> - Complex[0.01966912300437454, 0.], {21, 23} -> - Complex[1.1079404294454619`*^-8, -0.0003316699566825008], {21, 24} -> - Complex[-1.107940429399415*^-8, 2.241454781417565*^-19], {22, 23} 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78, 0, 70, "FooterCell",ExpressionUUID->"ba1c4e64-af83-4120-9883-a4a0a883d602"] +Cell[140536, 5288, 4964, 134, 70, "SeeAlsoSection",ExpressionUUID->"43d169e0-aab0-4946-aa98-aad05336b5a4"], +Cell[145503, 5424, 1792, 46, 70, "TechNotesSection",ExpressionUUID->"5a940d5b-584d-4c6c-b85e-c7ae76ce3025"], +Cell[147298, 5472, 1752, 45, 70, "MoreAboutSection",ExpressionUUID->"84e6bac4-bf42-449a-a492-01ca423dac22"], +Cell[149053, 5519, 2018, 55, 70, "RelatedLinksSection",ExpressionUUID->"5555f279-d554-431b-b945-22e6e303e7d5"], +Cell[151074, 5576, 78, 0, 70, "FooterCell",ExpressionUUID->"a772a8dc-c32a-49fc-a073-2f33936985f7"] } ] *) diff --git a/Q3/Documentation/English/ReferencePages/Symbols/Zero.nb b/Q3/Documentation/English/ReferencePages/Symbols/Zero.nb index 088f48e27..7ce9d3086 100644 --- a/Q3/Documentation/English/ReferencePages/Symbols/Zero.nb +++ b/Q3/Documentation/English/ReferencePages/Symbols/Zero.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 34619, 867] -NotebookOptionsPosition[ 30285, 777] -NotebookOutlinePosition[ 32448, 832] -CellTagsIndexPosition[ 32363, 827] +NotebookDataLength[ 34804, 871] +NotebookOptionsPosition[ 30469, 781] +NotebookOutlinePosition[ 32630, 836] +CellTagsIndexPosition[ 32545, 831] WindowTitle->Zero WindowFrame->Normal*) @@ -27,11 +27,11 @@ Cell[BoxData[GridBox[{ TemplateBox[{12}, "Spacer1"], Cell["Q3 SYMBOL", "PacletNameCell", TextAlignment->Center,ExpressionUUID-> - "97f4799e-69ea-4567-83aa-1546e648e463"], + "cf2403da-3083-495d-839d-9d5e9ce52740"], TemplateBox[{8}, "Spacer1"]}]], TextAlignment->Center,ExpressionUUID-> - "2d64bb67-985b-49b1-b929-ac90e0c64aad"], + "b3ecd590-2247-4a17-9ff4-3782957b3477"], Background->RGBColor[0.490196, 0.576471, 0.690196], ItemSize->Full], ""} }, @@ -54,8 +54,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "7b5fc3d2-e28f-4785-9ed0-147e10addcce"] - }],ExpressionUUID->"e77250e6-5bef-4a03-9dd5-45e2520761d3"], + "2e0f897a-01b3-4db7-814b-87894eb8df77"] + }],ExpressionUUID->"33615118-33cf-4ce5-a208-624579b2bb2c"], StripOnInput->False],{ "\"ConstantArray\"" :> Documentation`HelpLookup["paclet:ref/ConstantArray"], @@ -72,7 +72,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"SeeAlso"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "3715c4b9-832b-4d12-a864-18ab8f2b9d97"], + "586e6b38-65d8-418e-b7a1-0eb6c00b854d"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -91,8 +91,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "8cb10dc8-6f26-4825-9069-dad532e11e76"] - }],ExpressionUUID->"71818ecb-e511-4205-abc7-b10f7c8ea1fc"], + "44f11093-7854-4cb0-b790-bade27f71cc5"] + }],ExpressionUUID->"e40abb0d-5982-4836-a462-2b532826302c"], StripOnInput->False],{ "\"Quantum Information Systems\"" :> Documentation`HelpLookup[ @@ -107,7 +107,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"MoreAbout"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "deb78c42-4c06-4973-9fd1-759f13ec8046"], + "15bd3ba4-d0e4-4307-8277-da9e56a771cd"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -126,8 +126,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "5133eb87-f2e5-49d3-a2a7-c62a6d59cb5e"] - }],ExpressionUUID->"98538d46-adb3-4606-acb7-164f14ec94af"], + "7d48393e-5eb2-4c0d-9c2c-a83231e29298"] + }],ExpressionUUID->"ae341faa-6fff-4b0a-9b26-1cbebb9453db"], StripOnInput->False],{ "\"Quantum Information Systems with Q3\"" :> Documentation`HelpLookup[ @@ -144,7 +144,7 @@ Cell[BoxData[GridBox[{ MenuStyle->"Tutorials"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "cafe5d5d-c9ea-4cd9-843e-e673e60ba021"], + "83718659-89e7-4583-9dce-7f54e366d639"], "\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", Cell[BoxData[ @@ -163,8 +163,8 @@ Cell[BoxData[GridBox[{ ImageSize->20, PlotRange->{{-3, 4}, {-1, 1}}, PlotRangePadding->Scaled[0.025]]],ExpressionUUID-> - "4241127b-054f-4a3f-8012-9539639b519a"] - }],ExpressionUUID->"5d208a7a-b4dd-4d49-a442-7347c5fda57c"], + "cf8ac6f2-2a0f-4a42-b6ff-627695e0cd75"] + }],ExpressionUUID->"3427e016-12c1-46b8-bb90-c1cdd9d2b8fd"], StripOnInput->False],{ "\"Q3/ref/Zero\"" :> None, "\"Copy Wolfram Documentation Center URL\"" :> @@ -200,14 +200,14 @@ Cell[BoxData[GridBox[{ MenuStyle->"URLMenu"], MouseAppearanceTag["LinkHand"]]], LineSpacing->{1.4, 0},ExpressionUUID-> - "5babae97-1718-459f-beb5-068738b9b5de"] + "a2e164db-c503-4564-a14a-9927e7a52ad2"] }], "AnchorBar", CacheGraphics->False,ExpressionUUID-> - "cc081a39-b9e5-4a5c-a3eb-d73bb11e7fcf"]} + "240dadc1-a679-4b0c-9c6d-54bf9efd7e02"]} }]], "AnchorBarGrid", - CellID->1,ExpressionUUID->"1fb82a96-b009-4abd-9aa8-016a75ac2d60"], + CellID->1,ExpressionUUID->"fb772d4b-916f-4581-b30c-3c8f1402e5d8"], -Cell["Q3`", "ContextNameCell",ExpressionUUID->"71dddf70-2758-4a22-b01f-42daf1f23564"], +Cell["Q3`", "ContextNameCell",ExpressionUUID->"bffcbc78-9fd6-4e05-9496-38592e4bcaf5"], Cell[CellGroupData[{ @@ -215,17 +215,34 @@ Cell[BoxData[GridBox[{ {Cell[TextData[{ Cell[ "Zero", "ObjectName",ExpressionUUID-> - "ed18f385-c1d2-42a3-8993-36b25db3f999"], + "ed0cd773-2303-41b9-9b3c-f8711a142be5"], Cell[BoxData[ TemplateBox[{8}, - "Spacer1"]],ExpressionUUID->"c4feb301-5624-43ee-a865-7cb97a5ed957"], + "Spacer1"]],ExpressionUUID->"0461f284-2e0c-47f5-8126-d666b9ed6a48"], Cell[BoxData[ ""], "ObjectNameTranslation",ExpressionUUID-> - "da8146b5-fd23-4e95-864e-d3c8da5856e2"] - }],ExpressionUUID->"f2fc2d24-31c5-4e32-8f67-830c2aa7cb07"], - "\[SpanFromLeft]"} - }]], "ObjectNameGrid",ExpressionUUID->"e2636296-421b-4947-8e1c-\ -96368a80a993"], + 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}]], "SeeAlsoSection",ExpressionUUID->"b1654806-e2cc-40f0-95bc-\ -a2658546cc3b"], + }]], "SeeAlsoSection",ExpressionUUID->"7c61aa9c-9eb8-4721-b8a0-\ +17712edd08d6"], Cell[BoxData[GridBox[{ { @@ -687,8 +691,8 @@ Cell[BoxData[GridBox[{ BaseStyle->{"Tutorials"}]], "Tutorials",ExpressionUUID-> "fcf097a2-c249-4166-b34f-bd0bf583091b"]}]} }]} - }]], "TechNotesSection",ExpressionUUID->"460b298c-142d-48cb-91a9-\ -7c9d5c2f61fc"], + }]], "TechNotesSection",ExpressionUUID->"3b59d9c9-e1bc-46d0-ac4d-\ +6e65421c7afa"], Cell[BoxData[GridBox[{ { @@ -706,7 +710,7 @@ Cell[BoxData[GridBox[{ "paclet:Q3/guide/QuantumInformationSystems"}, "RefLinkPlain", BaseStyle->{"MoreAbout"}]], "MoreAbout",ExpressionUUID-> - "766d1cf5-baa7-4b2b-b4fd-09d3ce805737"]}]}, + "50e0c99a-a348-482a-ba9c-7aa9093565ac"]}]}, { RowBox[{"\[FilledVerySmallSquare]", Cell[BoxData[ TemplateBox[{ @@ -715,7 +719,7 @@ Cell[BoxData[GridBox[{ "paclet:Q3/guide/QuantumManyBodySystems"}, "RefLinkPlain", BaseStyle->{"MoreAbout"}]], "MoreAbout",ExpressionUUID-> - "5f265307-a9e9-488d-bc1a-839a1ca9e638"]}]}, + "9a4bf12d-40f5-4af2-bfa8-15bc4dcad1a3"]}]}, { RowBox[{"\[FilledVerySmallSquare]", Cell[BoxData[ TemplateBox[{ @@ -724,7 +728,7 @@ Cell[BoxData[GridBox[{ "paclet:Q3/guide/QuantumSpinSystems"}, "RefLinkPlain", BaseStyle->{"MoreAbout"}]], "MoreAbout",ExpressionUUID-> - "1dc4b5e4-8b22-4a1f-ac80-cffb54a26cbf"]}]}, + "a14766a4-7dae-45a7-bd3b-e11165b96f0b"]}]}, { RowBox[{"\[FilledVerySmallSquare]", Cell[BoxData[ TemplateBox[{ @@ -732,10 +736,10 @@ Cell[BoxData[GridBox[{ TextData["Q3"]], "paclet:Q3/guide/Q3"}, "RefLinkPlain", BaseStyle->{"MoreAbout"}]], "MoreAbout",ExpressionUUID-> - "40f09709-d079-4104-a200-7703b049dee2"]}]} + "de05140d-9688-46e9-9ca4-839644315cd7"]}]} }]} - }]], "MoreAboutSection",ExpressionUUID->"de7149eb-a08f-41bf-a17e-\ -7d2243dbf020"], + }]], "MoreAboutSection",ExpressionUUID->"22292b94-cd1b-4598-a0ae-\ +cb392d7dbb9f"], Cell[BoxData[GridBox[{ { @@ -764,16 +768,16 @@ Cell[BoxData[GridBox[{ "https://doi.org/10.1007/978-3-030-91214-7"}, "WebLink", BaseStyle->{"RelatedLinks"}]],ExpressionUUID-> - 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Complex[-0.06484280119839642, -0.030698057141020495`], + Complex[-0.058184329771711826`, 0.07158727950915138], + Complex[-0.12356826959526161`, 0.1196420103156779], + Complex[0.12603821576152777`, 0.07294185738764525], + Complex[-0.051533703431992976`, 0.01783566097908106], + Complex[0.1198845298873601, -0.07069943197697807]}}, { + $CellContext`S[1, Q3`$], + $CellContext`S[2, Q3`$], + $CellContext`S[3, Q3`$]}, { + $CellContext`A[1, Q3`$]}], "Invisible" -> { + $CellContext`A[ + Rational[3, 2]]}]]],ExpressionUUID-> + "f1b50fd1-9d4c-45d8-9b15-c85301b90d92"], + "=", + Cell[BoxData[ + InterpretationBox[ + GraphicsBox[{ + {Dashing[{Small, Small}], LineBox[{}]}, + LineBox[{{{0, -1}, {4, -1}}, {{0, -3}, {4, -3}}, {{0, -4}, {4, -4}}, {{ + 0, -5}, {4, -5}}}], { + {GrayLevel[1], EdgeForm[Opacity[1]], + RectangleBox[{0.6, -5.4}, {1.4, -2.6}]}, InsetBox[ + StyleBox[ + TemplateBox[{"\"W\"", "\"\[Dagger]\""}, + "Superscript"], + StripOnInput->False, + FontSize->Scaled[0.1], + FontWeight->"Light", + FontSlant->Italic], {1., -4.}, NCache[ + ImageScaled[{Rational[1, 2], Rational[1, 2]}], + ImageScaled[{0.5, 0.5}]], Automatic, {1, 0}]}, { + LineBox[{{2, -3}, {2, -4}, {2, -5}, {2, -1}}], + {GrayLevel[1], EdgeForm[Opacity[1]], + DiskBox[{2, -3}, 0.09, + NCache[{Rational[1, 4] Pi, Rational[5, 4] Pi}, {0.7853981633974483, + 3.9269908169872414`}]], + {GrayLevel[0], + DiskBox[{2, -3}, 0.09, + NCache[{Rational[-3, 4] Pi, Rational[1, 4] Pi}, {-2.356194490192345, + 0.7853981633974483}]]}}, + {GrayLevel[1], EdgeForm[Opacity[1]], + DiskBox[{2, -4}, 0.09, + NCache[{Rational[1, 4] Pi, Rational[5, 4] Pi}, {0.7853981633974483, + 3.9269908169872414`}]], + {GrayLevel[0], + DiskBox[{2, -4}, 0.09, + NCache[{Rational[-3, 4] Pi, Rational[1, 4] Pi}, {-2.356194490192345, + 0.7853981633974483}]]}}, + {GrayLevel[1], EdgeForm[Opacity[1]], + DiskBox[{2, -5}, 0.09, + NCache[{Rational[1, 4] Pi, Rational[5, 4] Pi}, {0.7853981633974483, + 3.9269908169872414`}]], + {GrayLevel[0], + DiskBox[{2, -5}, 0.09, + 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Q3`QuantumCircuit[{ + Complex[-0.08138626657007564, 0.11060910476814076`] + + Complex[0.051733979926041264`, -0.050229390581745956`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 3]] + + Complex[0.09604042306637672, -0.10529510202990823`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 4]] + + Complex[-0.1388792947609741, 0.11417459649567595`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 5]] + + Complex[-0.128527581540391, -0.07900487598664005] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[3, 3]] + + Complex[-0.18016404005482634`, 0.14732655711035553`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[3, 4]] + + Complex[0.06870523474933359, -0.08401574569110887] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[3, 5]] + + Complex[0.14952760965108025`, 0.08045426084650668] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 3]] + + Complex[0.03557568721236744, 0.04521578337763854] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 4]] + + Complex[-0.19873073240773823`, 0.0938541906108053] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 5]] + + Complex[0.11558673369383032`, 0.002448427300174069] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[3, 3]] + + Complex[-0.005152767861884441, -0.0633386597941751] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[3, 4]] + + Complex[0.21638033759208553`, -0.08859795707436469] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[3, 5]] + + Complex[0.013837997573979272`, -0.1693840647617133] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 3]] + + Complex[0.06035805190248033, 0.055060829464049374`] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 4]] + + Complex[0.01026257233961623, 0.08384523078932155] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 5]] + + Complex[0.05158472994874559, 0.032508930037728714`] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[3, 3]] + + Complex[-0.08707682138782519, 0.00809569139414161] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[3, 4]] + + Complex[-0.4761415592069672, -0.29661192010228865`] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[3, 5]] + + Complex[0.08751773052364228, -0.019197348286855532`] Q3`Multiply[ + $CellContext`S[2, 3], + $CellContext`S[3, 3]] + + Complex[-0.10264813941127669`, 0.22745860481538333`] Q3`Multiply[ + $CellContext`S[2, 3], + $CellContext`S[3, 4]] + + Complex[-0.0760248287372839, -0.21814794963510997`] Q3`Multiply[ + $CellContext`S[2, 3], + $CellContext`S[3, 5]] + + Complex[-0.03029594158657436, 0.23045382905027395`] Q3`Multiply[ + $CellContext`S[2, 4], + $CellContext`S[3, 3]] + + Complex[-0.08415822828433797, -0.10265043185584594`] Q3`Multiply[ + $CellContext`S[2, 4], + $CellContext`S[3, 4]] + + Complex[-0.4448951632706247, 0.08870718098840288] Q3`Multiply[ + $CellContext`S[2, 4], + $CellContext`S[3, 5]] + + Complex[0.2024957721445796, -0.03954591927919972] Q3`Multiply[ + $CellContext`S[2, 5], + $CellContext`S[3, 3]] + + Complex[-0.313694213811919, -0.0383647845417856] Q3`Multiply[ + $CellContext`S[2, 5], + $CellContext`S[3, 4]] + + Complex[-0.06395293556993221, -0.07774104329906908] Q3`Multiply[ + $CellContext`S[2, 5], + $CellContext`S[3, 5]] + + Complex[-0.01484061518243892, 0.10485057227832838`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 3], + $CellContext`S[3, 3]] + + Complex[0.22326934005969606`, -0.008517159653754575] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 3], + $CellContext`S[3, 4]] + + Complex[0.03000493785652983, -0.11348084198191122`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 3], + $CellContext`S[3, 5]] + + Complex[-0.12009757384889805`, -0.14108190946241178`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 4], + $CellContext`S[3, 3]] + + Complex[0.1712602923918959, -0.30627817172102073`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 4], + $CellContext`S[3, 4]] + + Complex[0.017700577425703107`, 0.35648797062874726`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 4], + $CellContext`S[3, 5]] + + Complex[0.05406557547857658, -0.06120456827132027] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 5], + $CellContext`S[3, 3]] + + Complex[-0.20634766755694556`, -0.3428519711095273] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 5], + $CellContext`S[3, 4]] + + Complex[0.31305242985226855`, 0.18160826157449947`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 5], + $CellContext`S[3, 5]] + + Complex[0.024332523752541607`, 0.049475073166067376`] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 3], + $CellContext`S[3, 3]] + + Complex[-0.22304549255469858`, -0.04862419642810534] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 3], + $CellContext`S[3, 4]] + + Complex[-0.160864109403631, -0.17284755026700122`] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 3], + $CellContext`S[3, 5]] + + Complex[0.1635578730042193, -0.1720540128391752] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 4], + $CellContext`S[3, 3]] + + Complex[-0.4893924018797447, -0.12220213122012044`] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 4], + $CellContext`S[3, 4]] + + Complex[0.08593953020020266, 0.09225751143359792] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 4], + $CellContext`S[3, 5]] + + Complex[0.2064340921245885, 0.012637530625447126`] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 5], + $CellContext`S[3, 3]] + + Complex[0.014142039346924448`, 0.3804910465083136] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 5], + $CellContext`S[3, 4]] + + Complex[-0.22510327212714473`, -0.3780452221565024] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 5], + $CellContext`S[3, 5]] + + Complex[-0.019807212264532272`, -0.1382649038023435] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 3], + $CellContext`S[3, 3]] + + Complex[0.04081483570565463, 0.2278479870124981] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 3], + $CellContext`S[3, 4]] + + Complex[0.13612673736640346`, -0.034002845838632195`] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 3], + $CellContext`S[3, 5]] + + Complex[-0.012897472932767752`, -0.15992782239426395`] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 4], + $CellContext`S[3, 3]] + + Complex[-0.41061012950581666`, 0.08114464286973663] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 4], + $CellContext`S[3, 4]] + + Complex[-0.09655723340332145, 0.022858069087815673`] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 4], + $CellContext`S[3, 5]] + + Complex[0.029893271120490138`, -0.15190790187866965`] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 5], + $CellContext`S[3, 3]] + + Complex[0.28911275555003324`, 0.2165794949238844] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 5], + $CellContext`S[3, 4]] + + Complex[-0.3470754944838056, 0.17619499870477603`] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 5], + $CellContext`S[3, 5]] + + Complex[0.061722774489510794`, -0.05881611102683626] $CellContext`S[ + 1, 3] + Complex[ + 0.25327341850529694`, -0.006785955325551724] $CellContext`S[1, 4] + + Complex[0.01091398072559055, -0.14752558722195544`] $CellContext`S[1, 5] + + Complex[-0.12727498132613876`, 0.017140100378421678`] $CellContext`S[ + 2, 3] + Complex[ + 0.03348080973588923, 0.05307841482712171] $CellContext`S[2, 4] + + Complex[-0.04372663565919723, -0.09590892542406253] $CellContext`S[2, 5] + + Complex[-0.06299766198284856, 0.05631962485500627] $CellContext`S[3, 3] + + Complex[-0.1572697248194979, -0.16936991685682976`] $CellContext`S[3, 4] + + Complex[0.10290083350689266`, -0.1155033758802036] $CellContext`S[3, 5], + "Label" -> Superscript["W", "\[Dagger]"]}, + Q3`UniformlyControlledGate[{ + $CellContext`S[1, Q3`$], + $CellContext`S[2, Q3`$], + $CellContext`S[3, Q3`$]}, { + 0.8660254037844387 $CellContext`A[1, 1] + + 0.4999999999999999 $CellContext`A[1, 3], + 0.9006814854556114 $CellContext`A[1, 1] + + 0.4344799900541716 $CellContext`A[1, 3], + 0.9182443284910992 $CellContext`A[1, 1] + + 0.39601433458137647` $CellContext`A[1, 3], + 0.950317707997143 $CellContext`A[1, 1] + + 0.3112816311108909 $CellContext`A[1, 3], + 0.9769856114869457 $CellContext`A[1, 1] + + 0.21330521547181816` $CellContext`A[1, 3], + 0.9865417387710291 $CellContext`A[1, 1] + + 0.16350962559627658` $CellContext`A[1, 3], + 0.996382903255763 $CellContext`A[1, 1] + + 0.0849771151523568 $CellContext`A[1, 3], + 0.9994649964844936 $CellContext`A[1, 1] + + 0.03270658652704571 $CellContext`A[1, 3]}, "Label" -> "G"], { + Complex[0.0027164756724508063`, -0.10539259404541908`] + + Complex[0.014829066432302328`, 0.002952909384853139] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 3]] + + Complex[0.04137271522158836, 0.15489738518322008`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 4]] + + Complex[0.09244507595299983, -0.035647964979481944`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 5]] + + Complex[0.1594185614107827, 0.004064152705538171] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[3, 3]] + + Complex[-0.11002616221121048`, 0.062316401272460666`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[3, 4]] + + Complex[0.1870743587101656, -0.05291328941715277] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[3, 5]] + + Complex[0.12136971744895858`, -0.08420825587764712] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 3]] + + Complex[0.01245940896323025, -0.0018517906662471782`] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 4]] + + Complex[0.15887893759125404`, 0.007969603794872781] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 5]] + + Complex[-0.07410080891211049, 0.08790249518935771] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[3, 3]] + + Complex[-0.07963132478987046, -0.12071176780618248`] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[3, 4]] + + Complex[0.17106682434381643`, 0.23781953932526556`] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[3, 5]] + + Complex[0.15369660940613294`, 0.13265790736079403`] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 3]] + + Complex[0.03661311659583397, -0.21011599249661284`] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 4]] + + Complex[0.056238283622095604`, -0.10813717522161084`] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 5]] + + Complex[-0.08896363608223251, 0.01708812046590201] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[3, 3]] + + Complex[-0.08683945294492326, -0.1231658093687507] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[3, 4]] + + Complex[0.37422858510783563`, -0.253885836375516] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[3, 5]] + + Complex[0.13608699304692362`, -0.04758965462608312] Q3`Multiply[ + $CellContext`S[2, 3], + $CellContext`S[3, 3]] + + Complex[-0.04889470935437945, -0.13550451260892646`] Q3`Multiply[ + $CellContext`S[2, 3], + $CellContext`S[3, 4]] + + Complex[0.034791171515207414`, 0.12065481266070595`] Q3`Multiply[ + $CellContext`S[2, 3], + $CellContext`S[3, 5]] + + Complex[0.11418726492756276`, -0.11537543896340599`] Q3`Multiply[ + $CellContext`S[2, 4], + $CellContext`S[3, 3]] + + Complex[0.0890500621487427, -0.14358258948850344`] Q3`Multiply[ + $CellContext`S[2, 4], + $CellContext`S[3, 4]] + + Complex[0.14786447048568804`, 0.0606872011073626] Q3`Multiply[ + $CellContext`S[2, 4], + $CellContext`S[3, 5]] + + Complex[-0.08232887972257927, 0.19182620543726692`] Q3`Multiply[ + $CellContext`S[2, 5], + $CellContext`S[3, 3]] + + Complex[0.27639397226525736`, -0.2039171093478583] Q3`Multiply[ + $CellContext`S[2, 5], + $CellContext`S[3, 4]] + + Complex[0.116999294259227, -0.04763785708520277] Q3`Multiply[ + $CellContext`S[2, 5], + $CellContext`S[3, 5]] + + Complex[-0.09813784922195307, -0.04002748247056523] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 3], + $CellContext`S[3, 3]] + + Complex[-0.16961282172759173`, -0.02495373053823364] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 3], + $CellContext`S[3, 4]] + + Complex[-0.15204562338428404`, -0.05123777308717774] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 3], + $CellContext`S[3, 5]] + + Complex[0.06984990356331504, -0.10057293239731374`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 4], + $CellContext`S[3, 3]] + + Complex[-0.26882362125094894`, 0.14358258948850344`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 4], + $CellContext`S[3, 4]] + + Complex[0.03393180056272981, 0.11811471999325622`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 4], + $CellContext`S[3, 5]] + + Complex[-0.07861956206905496, -0.2050020943144475] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 5], + $CellContext`S[3, 3]] + + Complex[0.08465464103679622, -0.19094694893885975`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 5], + $CellContext`S[3, 4]] + + Complex[0.17505014165759109`, -0.20743053921755064`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 5], + $CellContext`S[3, 5]] + + Complex[0.07338589585414926, 0.002861123084542673] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 3], + $CellContext`S[3, 3]] + + Complex[-0.23355978120691062`, 0.12071176780618248`] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 3], + $CellContext`S[3, 4]] + + Complex[0.32854815142312555`, -0.05735731421786801] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 3], + $CellContext`S[3, 5]] + + Complex[0.30234510436822637`, 0.0018517906662471782`] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 4], + $CellContext`S[3, 3]] + Complex[0.6787708058381466, 0.] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 4], + $CellContext`S[3, 4]] + + Complex[0.23279050399955822`, -0.4477086505370581] Q3`Multiply[ + $CellContext`S[1, 4], + 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0.08384523078932155] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[2, 5]] + + Complex[0.05158472994874559, 0.032508930037728714`] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[3, 3]] + + Complex[-0.08707682138782519, 0.00809569139414161] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[3, 4]] + + Complex[-0.4761415592069672, -0.29661192010228865`] Q3`Multiply[ + $CellContext`S[1, 5], + $CellContext`S[3, 5]] + + Complex[0.08751773052364228, -0.019197348286855532`] Q3`Multiply[ + $CellContext`S[2, 3], + $CellContext`S[3, 3]] + + Complex[-0.10264813941127669`, 0.22745860481538333`] Q3`Multiply[ + $CellContext`S[2, 3], + $CellContext`S[3, 4]] + + Complex[-0.0760248287372839, -0.21814794963510997`] Q3`Multiply[ + $CellContext`S[2, 3], + $CellContext`S[3, 5]] + + Complex[-0.03029594158657436, 0.23045382905027395`] Q3`Multiply[ + $CellContext`S[2, 4], + $CellContext`S[3, 3]] + + Complex[-0.08415822828433797, -0.10265043185584594`] Q3`Multiply[ + $CellContext`S[2, 4], + $CellContext`S[3, 4]] + + Complex[-0.4448951632706247, 0.08870718098840288] Q3`Multiply[ + $CellContext`S[2, 4], + $CellContext`S[3, 5]] + + Complex[0.2024957721445796, -0.03954591927919972] Q3`Multiply[ + $CellContext`S[2, 5], + $CellContext`S[3, 3]] + + Complex[-0.313694213811919, -0.0383647845417856] Q3`Multiply[ + $CellContext`S[2, 5], + $CellContext`S[3, 4]] + + Complex[-0.06395293556993221, -0.07774104329906908] Q3`Multiply[ + $CellContext`S[2, 5], + $CellContext`S[3, 5]] + + Complex[-0.01484061518243892, 0.10485057227832838`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 3], + $CellContext`S[3, 3]] + + Complex[0.22326934005969606`, -0.008517159653754575] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 3], + $CellContext`S[3, 4]] + + Complex[0.03000493785652983, -0.11348084198191122`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 3], + $CellContext`S[3, 5]] + + Complex[-0.12009757384889805`, -0.14108190946241178`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 4], + $CellContext`S[3, 3]] + + Complex[0.1712602923918959, -0.30627817172102073`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 4], + $CellContext`S[3, 4]] + + Complex[0.017700577425703107`, 0.35648797062874726`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 4], + $CellContext`S[3, 5]] + + Complex[0.05406557547857658, -0.06120456827132027] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 5], + $CellContext`S[3, 3]] + + Complex[-0.20634766755694556`, -0.3428519711095273] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 5], + $CellContext`S[3, 4]] + + Complex[0.31305242985226855`, 0.18160826157449947`] Q3`Multiply[ + $CellContext`S[1, 3], + $CellContext`S[2, 5], + $CellContext`S[3, 5]] + + Complex[0.024332523752541607`, 0.049475073166067376`] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 3], + $CellContext`S[3, 3]] + + Complex[-0.22304549255469858`, -0.04862419642810534] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 3], + $CellContext`S[3, 4]] + + Complex[-0.160864109403631, -0.17284755026700122`] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 3], + $CellContext`S[3, 5]] + + Complex[0.1635578730042193, -0.1720540128391752] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 4], + $CellContext`S[3, 3]] + + Complex[-0.4893924018797447, -0.12220213122012044`] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 4], + $CellContext`S[3, 4]] + + Complex[0.08593953020020266, 0.09225751143359792] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 4], + $CellContext`S[3, 5]] + + Complex[0.2064340921245885, 0.012637530625447126`] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 5], + $CellContext`S[3, 3]] + + Complex[0.014142039346924448`, 0.3804910465083136] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 5], + $CellContext`S[3, 4]] + + Complex[-0.22510327212714473`, -0.3780452221565024] Q3`Multiply[ + $CellContext`S[1, 4], + $CellContext`S[2, 5], 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Equivalent quantum circuits representing the GSLW block encoding of \ +matrix ", + Cell[BoxData[ + FormBox["M", TraditionalForm]],ExpressionUUID-> + "9779b073-0671-4ffb-9b8a-464803e91d90"], + " that has singular value decomposition ", + Cell[BoxData[ + FormBox[ + RowBox[{"M", "=", + RowBox[{"V", "\[VeryThinSpace]", "\[CapitalLambda]", "\[VeryThinSpace]", + SuperscriptBox["W", "\[Dagger]"]}]}], TraditionalForm]],ExpressionUUID-> + "b54bcf24-eb73-4be1-ba1b-1673424bd072"], + " with ", + Cell[BoxData[ + FormBox[ + RowBox[{"\[CapitalLambda]", ":=", + RowBox[{"diag", "(", + RowBox[{ + RowBox[{ + SubscriptBox["\[Lambda]", "1"], ",", + SubscriptBox["\[Lambda]", "2"]}], "\[InvisibleComma]", + RowBox[{",", "\[Ellipsis]", ",", + SubscriptBox["\[Lambda]", "n"]}]}], ")"}]}], TraditionalForm]], + ExpressionUUID->"ae369396-af0b-40de-8c81-eb238f1901d7"], + ". Here, ", + Cell[BoxData[ + FormBox[ + SubscriptBox["G", "k"], TraditionalForm]],ExpressionUUID-> + "a78a16ef-12a2-4b25-9c8c-9f58fe2c4570"], + " is a short-hand notation for ", + Cell[BoxData[ + FormBox[ + RowBox[{ + SubscriptBox["G", "k"], ":=", + RowBox[{ + RowBox[{ + SubscriptBox["G", "x"], "(", + SubscriptBox["\[Lambda]", "k"], ")"}], "=", + RowBox[{ + SubscriptBox["G", "x"], "(", + SubscriptBox["cos\[Theta]", "k"], ")"}]}]}], TraditionalForm]], + ExpressionUUID->"5f34dc7d-5a31-4fc7-8a57-aa5587936fab"], + " with ", + Cell[BoxData[ + FormBox[ + RowBox[{ + RowBox[{"cos", "(", + SubscriptBox["\[Theta]", "k"], ")"}], ":=", + SubscriptBox["\[Lambda]", "k"]}], TraditionalForm]],ExpressionUUID-> + "63a9be62-9c8a-412d-958a-dcdb0dcffbed"], + "." +}], "TechNoteCaption", + CellID->1045391992,ExpressionUUID->"b22f7c6b-e94c-4788-b184-ae0dea0989a5"] +}, Open ]], + +Cell[CellGroupData[{ + +Cell["Geometric View: Relation to the Grover Rotation", "Subsection", + CellID->1339493437,ExpressionUUID->"2c996f7e-877b-4bd9-949d-98da0139f1fe"], + +Cell["\<\ +The GSLW form of block encoding also allows a similar geometric \ +interpretation as the above. To see this, let us introduce the rotated basis \ +vector\ +\>", "TechNoteText", + CellID->795124158,ExpressionUUID->"fb1e2dc8-9180-479b-a5ed-c95e9466c68c"], + +Cell[TextData[{ + "\t", + Cell[BoxData[ + FormBox[ + RowBox[{ + TemplateBox[{ + RowBox[{"x", ":", + RowBox[{"-", "\[Theta]"}]}]}, + "Ket"], ":=", + RowBox[{ + RowBox[{ + RowBox[{ + SubsuperscriptBox["R", "x", "\[Dagger]"], "(", "\[Theta]", ")"}], + TemplateBox[{"0"}, + "Ket"]}], "\[DotEqual]", + RowBox[{"(", GridBox[{ + { + RowBox[{"cos", "(", + RowBox[{"\[Theta]", "/", "2"}], ")"}]}, + { + RowBox[{"\[ImaginaryI]", "\[VeryThinSpace]", + RowBox[{"sin", "(", + RowBox[{"\[Theta]", "/", "2"}], ")"}]}]} + }, + GridBoxAlignment->{"Columns" -> {{Right}}, "Rows" -> {{Baseline}}}], + ")"}]}]}], TraditionalForm]],ExpressionUUID-> + "ba5fc611-417b-4fe3-a05d-993bfa97a8bf"], + " " +}], "TechNoteText", + CellID->2008471872,ExpressionUUID->"33e2169c-db34-48b7-b587-b3e6fa8cee88"], + +Cell[TextData[{ + "and the ", + StyleBox["reflection", + 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"All",ExpressionUUID->"19a4019f-058c-46d6-8dc0-2f056ccd88c5"], +Cell[2418, 73, 100, 1, 70, 27, 0, "StyleData", "Outline3", "All",ExpressionUUID->"2137d07f-cc21-4299-b30d-ba010d62ea13"], +Cell[2521, 76, 100, 1, 70, 27, 0, "StyleData", "Outline4", "All",ExpressionUUID->"4167d299-e280-49b1-aae2-11afa96a0f48"] } ] *) diff --git a/Q3/Kernel/Abby.wl b/Q3/Kernel/Abby.wl index ff7bf39ad..fed54f701 100644 --- a/Q3/Kernel/Abby.wl +++ b/Q3/Kernel/Abby.wl @@ -26,6 +26,8 @@ BeginPackage["Q3`"] { LeftBrace, RightBrace, OverBrace, UnderBrace }; +{ ChebyshevCoefficients, ChebyshevSeries }; + { BinaryToGray, GrayToBinary, GrayToInteger, IntegerToGray, BitReflect, @@ -346,7 +348,7 @@ CountsFor[obj_List, var_List] := (**** ****) -IntegerParity::usage = "IntegerParity[n] returns 1 if n is even and -1 if n is odd." +IntegerParity::usage = "IntegerParity[n] returns 1 if n is an even integer and -1 if n is odd." SetAttributes[IntegerParity, Listable] @@ -604,6 +606,76 @@ ReplaceAllThrough[rules_][expr_] := ReplaceAllThrough[expr, rules] (***** *****) +(**** ****) + +ChebyshevCoefficients::usage = "ChebyshevCoefficients[func, n] returns Chebyshev coefficients c0, c1, c2, \[Ellipsis], cn that approximates function func with a polynomlial of degree n-1." + +ChebyshevCoefficients[fun_, d_Integer?NonNegative] := Module[ + { cc = Range[d + 1] - 1/2 }, + cc = Cos[cc*Pi/(d + 1)]; + cc = Map[fun, cc]; + cc = FourierDCT[cc, 2] * 2/Sqrt[d + 1]; + cc[[1]] /= 2; + cc +] + +(**** ****) + + +(**** ****) + +ChebyshevSeries::usage = "ChebyshevSeries[{c0, c2, c4, \[Ellipsis]}, 1] represents the series in Chebysehv polynomlials of even parity with coefficients {c0, c2, c4, \[Ellipsis]}.\nChebyshevSeries[{c1, c3, c5, \[Ellipsis]}, -1] represents the series in Chebysehv polynomlials of odd parity with coefficients {c1, c3, c5, \[Ellipsis]}.\nCChebyshevSeries[{c0, c1, c2, \[Ellipsis]}, 0] represents the series in Chebysehv polynomlials with coefficients {c0, c1, c2, \[Ellipsis]}.\nChebyshevSeries[{c0, c1, c2, \[Ellipsis]}] is automatically converted to ChebyshevSeries[coeff, parity] according to the parity indicated in {c0, c1, c2, \[Ellipsis]}." + +ChebyshevSeries /: +MakeBoxes[func:ChebyshevSeries[cc_?VectorQ, parity:(-1|0|1)], fmt_] := + BoxForm`ArrangeSummaryBox[ + ChebyshevSeries, func, None, + { BoxForm`SummaryItem @ { "Degree: ", func["Degree"] }, + BoxForm`SummaryItem @ { "Parity: ", Switch[parity, 1, "Even", -1, "Odd", _, "Indefinite"] } + }, + { BoxForm`SummaryItem @ { "Coefficients: ", cc[[;;UpTo[20]]] } + }, + fmt, + "Interpretable" -> Automatic + ] + +ChebyshevSeries[cc_?VectorQ, parity:(-1|1|0)]["Degree"] := + Switch[ parity, + +1, 2 * Length[cc] - 2, + -1, 2 * Length[cc] - 1, + -0, Length[cc] - 1 + ] + + +ChebyshevSeries[spec__][xx_List] := + Map[ChebyshevSeries[spec], xx] + + +ChebyshevSeries[cc_?VectorQ, 0][x_] := + Dot[cc, ChebyshevT[Range[Length @ cc] - 1, x]] + +ChebyshevSeries[cc_?VectorQ, 1][x_] := + Dot[cc, ChebyshevT[2*Range[Length @ cc] - 2, x]] + +ChebyshevSeries[cc_?VectorQ, -1][x_] := + Dot[cc, ChebyshevT[2*Range[Length @ cc] - 1, x]] + + +ChebyshevSeries[{c_}] := ChebyshevSeries[{c}, 1] + +ChebyshevSeries[cc_?VectorQ] := With[ + { aa = cc[[1;; ;;2]], + bb = cc[[2;; ;;2]] }, + Which[ + ZeroQ @ Norm[N @ aa, 1], ChebyshevSeries[bb, -1], + ZeroQ @ Norm[N @ bb, 1], ChebyshevSeries[aa, +1], + True, ChebyshevSeries[cc, 0] + ] +] + +(**** ****) + + (**** ****) LeftBrace::usage = "LeftBrace[y] returns the left brace spanning from -y to y." diff --git a/Q3/Kernel/Abel.wl b/Q3/Kernel/Abel.wl index 7591f790d..0f47a4f62 100644 --- a/Q3/Kernel/Abel.wl +++ b/Q3/Kernel/Abel.wl @@ -78,11 +78,11 @@ $::usage = "$ is a flavor index referring to the species itself." { PauliGroup, FullPauliGroup, PauliGroupElements, FullPauliGroupElements, - PauliQ }; + PauliQ, PauliMatrixQ }; { CliffordGroup, FullCliffordGroup, CliffordGroupElements, FullCliffordGroupElements, - CliffordQ }; + CliffordQ, CliffordMatrixQ }; (**** ****) @@ -1067,7 +1067,7 @@ AddElaborationPatterns[_MultiplyExp]; AddElaborationPatterns[ Exp[a_] :> MultiplyExp[a] /; Not[FreeQ[a, _?NonCommutativeQ]] - ] +] Format @ HoldPattern @ MultiplyExp[expr_] := diff --git a/Q3/Kernel/Dicke.wl b/Q3/Kernel/Dicke.wl index abee6f334..4098dab85 100644 --- a/Q3/Kernel/Dicke.wl +++ b/Q3/Kernel/Dicke.wl @@ -5,29 +5,34 @@ BeginPackage["Q3`"] Begin["`Private`"] -DickeBasis::usage = "DickeBasis[{S1, S2, ...}, {b1, b2, ...}, {m,n}] constructs a basis for Qubits/Qudits S1, S2, ... and Bosonic modes b1, b2, ..., where the number of particles in each Bosonic modes is limited in the range [m, n].\nDickeBasis[ {S1, S2, ...}, {b1, b2, ...}, n] is equivalent to DickeBasis[ {S1, S2, ...}, {b1, b2, ...}, {0,n}]." - -DickeBasis[ s_?QubitQ, b_?BosonQ, n_Integer ] := - DickeBasis[{s}, {b}, n] - -DickeBasis[ ss:{__?QubitQ}, b_?BosonQ, n_Integer ] := - DickeBasis[ss, {b}, n] - -DickeBasis[ s_?QubitQ, bb:{__?BosonQ}, n_Integer ] := - DickeBasis[{s}, bb, n] - -DickeBasis[ ss:{__?QubitQ}, bb:{__?BosonQ}, n_Integer ] := - DickeBasis[ ss, bb, {0, n} ] - -DickeBasis[ ss:{__?QubitQ}, bb:{__?BosonQ}, {m_Integer, n_Integer} ] := - Flatten @ Outer[ CircleTimes, Basis @ ss, BosonBasis[bb, {m,n}] ] - - -DickeBasis[ ss:{__?QuditQ}, bb:{__?BosonQ}, n_Integer ] := - DickeBasis[ ss, bb, {0, n} ] - -DickeBasis[ ss:{__?QuditQ}, bb:{__?BosonQ}, {m_Integer, n_Integer} ] := - Flatten @ Outer[ CircleTimes, Basis @ ss, BosonBasis[bb, {m,n}] ] +(**** ****) + +DickeBasis::usage = "DickeBasis[s1, s2, \[Ellipsis], {m, n}] returns the computational basis for a system Flatten[{s1, s2, \[Ellipsis]}] of qubits, qudits, bosons or a list of such species, where the number of particles in each bosonic mode is limited in Range[m, n].\nDickeBasis[spec, n] is equivalent DickeBasis[spec, {m, n}], where m is the maximum of the Bottom values of the bosonic moodes.\nDickeBasis[spec] is equivalent to DickeBasis[spec, {m, n}], where m is the maximum of the Bottom values of the bosonic moodes and n is the minimum of the Top values of the bosonic modes." + +DickeBasis[ss:(_?QubitQ | _?QuditQ | _?BosonQ | {(_?QubitQ | _?QuditQ | _?BosonQ)..})..] := + Module[ + { bb = Cases[Flatten @ {ss}, _?BosonQ], + m, n }, + m = Max[Bottom /@ bb]; + n = Min[Top /@ bb]; + Flatten @ Outer[CircleTimes, Basis @ Supplement[Flatten @ {ss}, bb], BosonBasis[bb, {m, n}]] + ] + +DickeBasis[ss:(_?QubitQ | _?QuditQ | _?BosonQ | {(_?QubitQ | _?QuditQ | _?BosonQ)..}).., n_Integer] := + Module[ + { bb = Cases[Flatten @ {ss}, _?BosonQ], + m }, + m = Max[Bottom /@ bb]; + Flatten @ Outer[CircleTimes, Basis @ Supplement[Flatten @ {ss}, bb], BosonBasis[bb, {m, n}]] + ] + +DickeBasis[ss:(_?QubitQ | _?QuditQ | _?BosonQ | {(_?QubitQ | _?QuditQ | _?BosonQ)..}).., {m_Integer, n_Integer}] := + With[ + { bb = Cases[Flatten @ {ss}, _?BosonQ] }, + Flatten @ Outer[CircleTimes, Basis @ Supplement[Flatten @ {ss}, bb], BosonBasis[bb, {m, n}]] + ] + +(**** ****) End[] diff --git a/Q3/Kernel/Gottesman.wl b/Q3/Kernel/Gottesman.wl index c6e85e505..4111925ac 100644 --- a/Q3/Kernel/Gottesman.wl +++ b/Q3/Kernel/Gottesman.wl @@ -160,16 +160,16 @@ GroupMultiplicationTable @ PauliGroup[ss:{__?QubitQ}] := (**** ****) -(**** ****) +(**** ****) -PauliQ::usage = "PauliQ[op] returns True if operator op is an element of the full Pauli group." +PauliMatrixQ::usage = "PauliMatrixQ[mat] returns True if matrix mat is an element of the full Pauli group, and False otherwise." (* NOTE: The previous method based on PauliDecompose is slow for a large number of qubits, especially, 8 or more qubits. *) -PauliQ[mat_?SquareMatrixQ] := thePauliQ[mat] /; Length[mat] == 2 +PauliMatrixQ[mat_?SquareMatrixQ] := thePauliMatrixQ[mat] /; Length[mat] == 2 -PauliQ[mat_?SquareMatrixQ] := Module[ +PauliMatrixQ[mat_?SquareMatrixQ] := Module[ { tsr = Tensorize[mat], dim, kk, tt, n }, dim = Dimensions[tsr]; @@ -177,49 +177,72 @@ PauliQ[mat_?SquareMatrixQ] := Module[ kk = NestList[RotateLeft[#, 2]&, Range[2*n], n-1]; tt = Transpose[tsr, #]& /@ kk; tt = Map[ - Function[{x}, Flatten @ Map[thePauliQ, x, {2*(n-1)}]], - tt ]; + Function[{x}, Flatten @ Map[thePauliMatrixQ, x, {2*(n-1)}]], + tt + ]; If[ AllTrue[tt, (Length[#] == Power[2, n-1])&], AllTrue[Flatten @ tt, Identity], - False ] - ] /; IntegerQ @ Log[2, Length @ mat] /; Length[mat] > 2 + False + ] +] /; IntegerQ @ Log[2, Length @ mat] /; Length[mat] > 2 + +PauliMatrixQ[_List] = False + +PauliMatrixQ[_Association] = False + + +thePauliMatrixQ::usage = "thePauliMatrixQ[mat] returns True if 2x2 matrix mat is one of the Pauli operators." + +thePauliMatrixQ[mat_?SquareMatrixQ] := Module[ + { cc }, + cc = Chop @ { + Tr @ mat, + Tr @ Reverse @ mat, + Subtract @@ Diagonal @ mat, + Subtract @@ Diagonal @ Reverse @ mat + }; + cc = DeleteCases[cc, 0]; + Switch[ Length[cc], + 0, Nothing, + 1, True, + _, False + ] +] /; Length[mat] == 2 + +thePauliMatrixQ[_] = False + +(**** ****) + -PauliQ[_List] = False +(**** ****) + +(* NOTE: PauliQ and PauliMatrixQ may be combined to a single function. However, we keep them separate because PauliMatrixQ requires rather a large computational cost for large matrices. *) + +PauliQ::usage = "PauliQ[op] returns True if operator op is an element of the full Pauli group, and False otherwise." -PauliQ[_Association] = False +SetAttributes[PauliQ, Listable] PauliQ[Pauli[{(0|1|2|3)..}]] = True -PauliQ[(-1|-I|I) * Pauli[{(0|1|2|3)..}]] = True +PauliQ[-1 | 1 | -I | I] = True PauliQ[_?QubitQ[___, 0|1|2|3]] = True -PauliQ[HoldPattern @ Multiply[(_?QubitQ[___, 0|1|2|3])..]] = True +PauliQ[HoldPattern @ Multiply[__?QubitQ]] = True -PauliQ[(-1|-I|I) * HoldPattern @ Multiply[(_?QubitQ[___, 0|1|2|3])..]] = True -PauliQ[expr_] := PauliQ[Matrix @ expr] +PauliQ[(-1|-I|I) * _?PauliQ] = True -thePauliQ::usage = "thePauliQ[mat] returns True if 2x2 matrix mat is one of the Pauli operators." +PauliQ[expr_Plus] := PauliQ[Elaborate @ expr] /; + Not[FreeQ[expr, _?QubitQ[___, (4|5|6|7|8|9|10|11)] | _Pauli[___, (4|5|6|7|8|9|10|11), ____]]] + +PauliQ[expr:HoldPattern[Times[__, _Plus]]] := PauliQ[Elaborate @ expr] /; -thePauliQ[mat_?SquareMatrixQ] := Module[ - {cc}, - cc = Chop @ { - Tr @ mat, - Tr @ Reverse @ mat, - Subtract @@ Diagonal @ mat, - Subtract @@ Diagonal @ Reverse @ mat }; - cc = DeleteCases[cc, 0]; - Switch[ Length[cc], - 0, Nothing, - 1, True, - _, False] - ] /; Length[mat] == 2 -thePauliQ[_] = False +PauliQ[_] = False (**** ****) @@ -339,31 +362,49 @@ CliffordGroupElements[spec:(_Integer|{__?QubitQ}), kk:{__Integer}] := ] -CliffordQ::usage = "CliffordQ[mat] returns True if matrix mat represents a Clifford operator; and False otherwise.\nCliffordQ[op] applies to operator op expressed in terms of the labelled or unlabelled Pauli operators." +(**** ****) + + +(**** ****) -CliffordQ::notqbt = "`` does not represent an operator acting on qubits." +CliffordMatrixQ::usage = "CliffordMatrixQ[mat] returns True if matrix mat represents a Clifford operator, and False otherwise." -CliffordQ[mat_?MatrixQ] := ( - Message[CliffordQ::notqbt, vec]; +CliffordMatrixQ::notqbt = "Matrix `` does not represent an operator acting on qubits." + + +CliffordMatrixQ[mat_?MatrixQ] := ( + Message[CliffordMatrixQ::notqbt, vec]; False - ) /; Not @ AllTrue[Log[2, Dimensions @ mat], IntegerQ] +) /; Not @ AllTrue[Log[2, Dimensions @ mat], IntegerQ] -CliffordQ[mat_?MatrixQ] := Module[ +CliffordMatrixQ[mat_?MatrixQ] := Module[ { n = Log[2, Length @ mat], spr = Supermap[mat], gnr }, gnr = ThePauli /@ Join[ NestList[RotateRight, PadRight[{1}, n], n-1], NestList[RotateRight, PadRight[{3}, n], n-1] - ]; - AllTrue[spr /@ gnr, PauliQ] - ] + ]; + AllTrue[spr /@ gnr, PauliMatrixQ] +] -CliffordQ[expr_] := CliffordQ[Matrix @ expr] +(**** ****) -CliffordQ[expr_, ss:{__?QubitQ}] := CliffordQ[Matrix @ expr] -(**** ****) +(**** ****) + +(* NOTE: CliffordQ and CliffordMatrixQ may be combined to a single function. We separate them in consistency with PauliQ and PauliMatrixQ. *) + +CliffordQ::usage = "CliffordQ[op] returns True if operator op is an element of the full Clifford group, and False otherwise." + +SetAttributes[CliffordQ, Listable] + + +CliffordQ[expr_] := CliffordMatrixQ[Matrix @ expr] + +CliffordQ[expr_, ss:{__?QubitQ}] := CliffordMatrixQ[Matrix @ expr] + +(**** ****) (**** ****) @@ -933,7 +974,7 @@ FromGottesmanMatrix[mat_?MatrixQ, ss:{_?QubitQ, __?QubitQ}] := Module[ opf = FromGottesmanMatrix[ff, qq]; - new = CirclePlus[ff, One[2(n-1)]]; + new = CirclePlus[ff, One[2*(n-1)]]; new = Mod[mat . new, 2]; aa = FromGottesmanVector[new[[2, 3;;]], rr]; @@ -1160,7 +1201,7 @@ CliffordFactor[mat_?MatrixQ, ss:{_?QubitQ, __?QubitQ}] := Module[ opf = Dagger @ FromGottesmanMatrix[ff, qq]; - new = CirclePlus[ff, One[2(n-1)]]; + new = CirclePlus[ff, One[2*(n-1)]]; new = Mod[mat . new, 2]; aa = FromGottesmanVector[new[[2, 3;;]], rr]; diff --git a/Q3/Kernel/Pauli.wl b/Q3/Kernel/Pauli.wl index 58db6f8c6..cbb91fef9 100644 --- a/Q3/Kernel/Pauli.wl +++ b/Q3/Kernel/Pauli.wl @@ -2144,13 +2144,16 @@ Matrix[ expr_, ss:{__?SpeciesQ}, tt:{__?SpeciesQ} ] := Matrix[expr, FlavorNone @ ss, FlavorNone @ tt] /; Not[FlavorNoneQ @ {ss, tt}] + +(* linearity *) + Matrix[ expr_Plus, qq:{___?SpeciesQ}.. ] := TrigToExp @ ExpToTrig @ With[ { new = KetChop @ expr }, If[Head[new] === Plus, Map[Matrix[#, qq]&, new], Matrix[new, qq]] - ] + ] (* NOTE: TrigToExp @ ExpToTrig helps simplify in many cases. *) -(* NOTE: KetChop is required here because "0. + Ket[...]" may happen. *) +(* NOTE: KetChop is required here because "0. + Ket[...] + ..." may happen. *) Matrix[ z_?CommutativeQ op_, qq:{___?SpeciesQ}.. ] := z * Matrix[op, qq] @@ -2160,12 +2163,13 @@ Matrix[ z_?CommutativeQ, qq:{__?SpeciesQ} ] := z * One[Times @@ Dimension @ qq] Matrix[ z_?CommutativeQ, ss:{__?SpeciesQ}, tt:{__?SpeciesQ} ] := - z * One[Times @@ Dimension @ ss, Times @@ Dimension @ tt] + z * One[{Times @@ Dimension @ ss, Times @@ Dimension @ tt}] + (* Dagger *) HoldPattern @ Matrix[Dagger[op_?AgentQ], rest___] := Topple @ Matrix[op, rest] -(* NOTE: Matrix[a] may still include some operators; and hence Topple instead +(* NOTE: Matrix[a] may still include some operators; hence, Topple instead of ConjugateTranspose. *) @@ -2180,8 +2184,8 @@ HoldPattern @ expr, {0 -> mat, else_ :> Matrix[else]}, {1} - ] - ] /; And[Not @ FreeQ[expr, _Pauli], MemberQ[expr, 0]] + ] + ] /; And[Not @ FreeQ[expr, _Pauli], MemberQ[expr, 0]] HoldPattern @ Matrix[expr_Association, ss:{___?SpeciesQ}..] := Map[Matrix[#, ss]&, expr] @@ -2199,7 +2203,6 @@ Matrix[Bra[any_], rest___] := Conjugate @ Matrix[Ket[any], rest] (* For Ket of unlabelled qubits *) - Matrix[ vec:Ket[_List], {___} ] := TheMatrix[vec] @@ -2220,13 +2223,12 @@ Matrix[ op:Pauli[_List], {___} ] := TheMatrix[op] (* For Fermions *) -Matrix[op_?FermionQ, qq:{__?SpeciesQ}] := fermionMatrix[op, qq] /; - MemberQ[FlavorNone @ qq, FlavorMute @ Peel @ op] +Matrix[op_?FermionQ, qq:{__?SpeciesQ}] := fermionMatrix[op, qq] +(* NOTE: Dagger[fermion] is handled by Matrix[Dagger[...], ...] above. *) (* NOTE: Matrix and MatrixIn must be consistent, recalling that Basis always sorts the Keys in Ket. This function also depends on Basis. *) - -fermionMatrix[op_, ff:{___?SpeciesQ}] := Module[ +fermionMatrix[op_?FermionQ, ff:{___?SpeciesQ}] := Module[ { mm }, mm = SequenceCases[ ff, {pre___, op, post___} :> { @@ -2236,14 +2238,14 @@ fermionMatrix[op_, ff:{___?SpeciesQ}] := Module[ ]; If[ mm == {}, Message[Matrix::fermion, op, ff]; - Return[TheMatirx @ op], - Return[CircleTimes @@ Join @@ First[mm]] + op * Matrix[1, ff], + CircleTimes @@ Join @@ First[mm] ] ] fermionOne[f_?FermionQ] := ThePauli[3] -fermionOne[any_] := One[Dimension @ any] +fermionOne[any_?SpeciesQ] := One[Dimension @ any] (* For other Species *) @@ -2255,22 +2257,23 @@ Matrix[op_?AnyAgentQ, ss:{__?SpeciesQ}] := Module[ rr = One /@ Dimension[ss]; rr = Association @ Join[ Thread[ss -> rr], {sp -> mm} ]; CircleTimes @@ rr - ] /; MemberQ[ss, FlavorMute @ Peel @ op] +] /; MemberQ[ss, FlavorMute @ Peel @ op] +(* NOTE: Dagger[op] is handled by Matrix[Dagger[...], ...] above, but other modifiers such as Canon may happen. *) Matrix[op_?AnyAgentQ, qq:{__?SpeciesQ}] := ( Message[Matrix::rmndr, op]; op * Matrix[1, qq] - ) +) (* For Dyad-like (but not Dyad) expression *) Matrix[ HoldPattern @ Multiply[pre___, ket_Ket, bra_Bra, post___], qq:{___?SpeciesQ} - ] := Dyad[ +] := Dyad[ Matrix[pre ** ket, qq], Dagger @ Matrix[bra ** post, qq] - ] +] (* NOTE: Dagger (not Conjugate) here. *) (* For Multiply[...] *) @@ -3365,13 +3368,13 @@ Dyad[a_] := Module[ { qq = Agents[a] }, Message[Dyad::one]; Dyad[a, a, qq] - ] /; Not @ FreeQ[a, _Ket] +] /; Not @ FreeQ[a, _Ket] Dyad[a_, b_] := Module[ { qq = Agents @ {a, b} }, Message[Dyad::two]; Dyad[a, b, qq] - ] /; Not @ FreeQ[{a, b}, Ket[_Association]] +] /; Not @ FreeQ[{a, b}, Ket[_Association]] (* Direct constuction of Dyad *) @@ -3597,34 +3600,47 @@ theDyadForm[val:{__}, spc:{__?SpeciesQ}] := Module[ (**** ****) -Zero::usage = "Zero[n] return an array of length n with all elements zero.\nZero[m, n, ...] \[Congruent] Zero[{m, n, ...}] returns an m x n x ... tensor with all elements zero." +Zero::usage = "Zero[n] return an one-diemnsional array of lenth n with all elements zero.\nZero[{m, n, ...}] returns an m\[Times]n\[Times]\[Ellipsis] tensor with all elements zero." -Zero[mn__Integer] := SparseArray[{}, {mn}] +Zero[n_Integer] := Zero[{n}] -Zero[{mn__Integer}] := SparseArray[{}, {mn}] +Zero[mn:{__Integer}] := SparseArray[{}, mn] + +Zero[m_Integer, n__Integer] := ( + Message[Q3General::changed, Zero, "Use the form Zero[{n1, n2, \[Ellipsis]}]."]; + Zero[{m, n}] +) (**** ****) (**** ****) -One::usage = "One[n] \[Congruent] One[{n}] is almost the same as IdentityMatrix[n], but returns the identity matrix in a sparse array.\nOne[m, n, ...] \[Congruent] One[{m, n}, ...] returns the m \[Times] n \[Times] ... pseudo-identity tensor, i.e., the tensor where the main diagonal elements are 1 and the other elements are all zero.\nOne[{m, n}, k] returns an m \[Times] n matrix with the elements on the \*SuperscriptBox[k,th] diagonal being 1 and zero elsewhere." - -One[] := One @ {2, 2} +One::usage = "One[n] or One[{n}] returns the n\[Times]n identiy matrix in a sparse array; cf. IdentityMatrix[n].\nOne[{m, n}, ...] returns the m\[Times]n\[Times]\[Ellipsis] pseudo-identity tensor, i.e., the tensor where the main diagonal elements are 1 and the other elements are all zero.\nOne[{m, n}, k] returns an m\[Times]n matrix with the elements on the \*SuperscriptBox[k,th] diagonal being 1 and zero elsewhere." One[n_Integer] := One @ {n, n} -One[m_Integer, n__Integer] := One @ {m, n} - One[{n_Integer}] := One @ {n, n} One[{m_Integer, n__Integer}] := - SparseArray[ ConstantArray[j_, Length @ {m, n}] -> 1, {m, n} ] + SparseArray[ConstantArray[j_, Length @ {m, n}] -> 1, {m, n}] + One[{n_Integer}, p_Integer] := One[{n, n}, p] -One[{m_Integer, n_Integer}, p_Integer] := - SparseArray[ {i_, j_} :> 1 /; j == i + p, {m, n} ] +One[{m_Integer, n_Integer}, k_Integer] := + SparseArray[{i_, j_} :> 1 /; j == i+k, {m, n}] + + +One[] := ( + Message[Q3General::changed, One, "Use One[2] or One[{2, 2}]."]; + One @ {2, 2} +) + +One[m_Integer, n__Integer] := ( + Message[Q3General::changed, One, "Use the form One[{n1, n2, \[Ellipsis]}]."]; + One @ {m, n} +) (**** ****) @@ -4420,7 +4436,7 @@ Purification[rho_] := With[ Purification[rho_] := ExpressionFor @ Purification @ Matrix[rho] /; Not @ FreeQ[rho, _Pauli] -Purification[z_?CommutativeQ] := ExpressionFor @ Purification[z*One[2]] +Purification[z_?CommutativeQ] := ExpressionFor @ Purification[z * One[2]] (* NOTE: Single qubit is assumed. *) (**** ****) @@ -4542,16 +4558,19 @@ RandomMatrix[range_, n_Integer] := RandomComplex[range, {n, n}] RandomMatrix[range_, mn:{_Integer, _Integer}] := RandomComplex[range, mn] +(**** ****) + RandomHermitian::usage = "RandomHermitian[n] gives a randomly generated n\[Times]n Hermitian matrix with each element in the range between -(1+I) and (1+I).\nRandomHermitian[] assumes n=2.\nRandomHermitian[range, n] gives a randomly generated n\[Times]n Hermitian matrix. The range specifies the range (see RandomComplex) of the elements." RandomHermitian[] := RandomHermitian[(1+I){-1, 1}, 2] RandomHermitian[n_Integer] := RandomHermitian[(1+I){-1, 1}, n] -RandomHermitian[range_, n_Integer] := With[ - { m = RandomComplex[range, {n, n}] }, - ( m + ConjugateTranspose[m] ) / 2 - ] +RandomHermitian[range_, n_Integer] := + Symmetrize[RandomComplex[range, {n, n}], Hermitian @ {1, 2}] + +(**** ****) + RandomPositive::usage = "RandomPositive[n] gives a randomly generated n\[Times]n positive matrix with each element in the range between -(1+I) and (1+I).\nRandomPositive[] assumes n=2.\nRandomPositive[range, n] gives a random positive matrix with each element in the specified 'range' (see RandomComplex for the details of the specification of range)." @@ -4562,7 +4581,12 @@ RandomPositive[n_Integer] := RandomPositive[(1+I){-1, 1}, n] RandomPositive[range_, n_Integer] := With[ { m = RandomMatrix[range, {n, n}] }, Topple[m] . m - ] +] + + +(**** ****) + +(* See Mezzadri (2007) *) RandomUnitary::usage = "RandomUnitary[n] generates a uniformly distributed random unitary matrix of dimension n. Here, the uniform distribution is in the sense of the Haar measure." @@ -4580,6 +4604,8 @@ RandomUnitary[n_Integer] := Module[ qq . rr ] +(**** ****) + (**** *****) diff --git a/Q3/Kernel/QML.wl b/Q3/Kernel/QML.wl index ba48cae5e..87ab19367 100644 --- a/Q3/Kernel/QML.wl +++ b/Q3/Kernel/QML.wl @@ -197,17 +197,34 @@ BlockEncoding[mat_?MatrixQ, ss:{__?QubitQ}, aa:{__?QubitQ}, opts___?OptionQ] := BlockEncoding /: -Expand @ BlockEncoding[mat_?MatrixQ, ss:{__?QubitQ}, aa:{__?QubitQ}, ___?OptionQ] := Module[ - { uu, dd, vv, mm }, +ExpandAll[enc:BlockEncoding[_?MatrixQ, {__?QubitQ}, {__?QubitQ}, ___?OptionQ]] := + Expand[Expand @ enc] + +BlockEncoding /: +Expand @ BlockEncoding[mat_?MatrixQ, ss:{__?QubitQ}, aa:{__?QubitQ}, opts___?OptionQ] := Module[ + { mint = {"V", "\[CapitalOmega]", "W"}, + text, uu, dd, vv }, {uu, dd, vv} = SingularValueDecomposition[mat]; dd = Diagonal[dd]; dd = dd * (Last[aa][3]) + Sqrt[1-dd^2] * (Last[aa][1]); uu = ExpressionFor[uu, ss]; vv = ExpressionFor[Topple @ vv, ss]; + + (* Handle the "Label" option. *) + text = OptionValue[ Gate, + Append[{opts}, "Label" -> {"M", mint}], + "Label" + ]; + If[ ListQ[text], + If[Length[text] > 1, text = Last[text], text = mint], + text = mint + ]; + If[Not[ListQ[text] && Length[text] >= 3], text = mint]; + QuantumCircuit[ - {vv, "Label" -> Superscript["W", "\[Dagger]"]}, - UniformlyControlledGate[ss, dd, "Label" -> "\[CapitalOmega]"], - {uu, "Label" -> "V"}, + {vv, "Label" -> Superscript[text[[3]], "\[Dagger]"]}, + UniformlyControlledGate[ss, dd, "Label" -> text[[2]], opts], + {uu, "Label" -> text[[1]]}, "Visible" -> aa ] ] @@ -223,7 +240,7 @@ Matrix @ BlockEncoding[mat_?MatrixQ, ss:{__?QubitQ}, aa:{__?QubitQ}, ___?OptionQ {mm, -dd} }; mm = CirclePlus[uu, uu] . dd . CirclePlus[Topple @ vv, Topple @ vv]; - CircleTimes[One[Power[2, Length[aa]-1]], mm] + CircleTimes[One @ Power[2, Length[aa]-1], mm] ] BlockEncoding /: @@ -241,13 +258,15 @@ ParseGate[ more___?OptionQ ] := Module[ - { new }, + { txt, new }, + txt = OptionValue[Gate, Append[{opts}, "Label" -> "M"], "Label"]; + If[ListQ[txt], txt = First[txt]]; new = FilterRules[ - { more, opts, + { "Label" -> txt, + more, opts, "ControlShape" -> "Oval", "ControlLabel" -> "enc", - "ControlLabelAngle" -> Pi/2, - "Label" -> "M" + "ControlLabelAngle" -> Pi/2 }, Options @ Gate ]; diff --git a/Q3/Kernel/QuantumCircuit.wl b/Q3/Kernel/QuantumCircuit.wl index 3c0e418ca..38427500d 100644 --- a/Q3/Kernel/QuantumCircuit.wl +++ b/Q3/Kernel/QuantumCircuit.wl @@ -154,7 +154,7 @@ QCAct[op_Measurement, post___] := QCAct[op:Except[_Measurement]..] := Elaborate @ Fold[ Garner[Multiply[#2, #1]]&, 1, {op} ] -(* NOtE: One can use Elaborate@{op} as follows: +(* NOtE: One can use Elaborate @ {op} as follows: Fold[ Garner[Multiply[#2, #1]]&, 1, Elaborate@{op} ] However, this cannot take the advantange of op ** Ket[...]. *) @@ -265,8 +265,8 @@ QuantumCircuit /: HoldPattern @ Graphics[ QuantumCircuit[gg__, opts___?OptionQ], more___?OptionQ ] := Module[ - { ss = Qubits@{gg}, - cc = ParseGate@{gg}, + { ss = Agents @ {gg}, + cc = ParseGate @ {gg}, vv, ww, xx, yy, in, out, unit, nodes, marks, lines }, {vv, ww, unit, $PortSize} = @@ -279,7 +279,7 @@ HoldPattern @ vv = FlavorNone @ Flatten @ doAssureList @ vv; ww = FlavorNone @ Flatten @ doAssureList @ ww; - ss = Union @ Flatten@{ss, vv, ww}; + ss = Union @ Flatten @ {ss, vv, ww}; cc = DeleteCases[cc, {}]; (* E.g., Measurement[{}] *) If[cc == {}, cc = {"Spacer"}]; (* E.g., only input elements *) @@ -296,10 +296,10 @@ HoldPattern @ marks = qcMarks @ Cases[{gg}, _Mark, Infinity]; - in = FirstCase[ {gg}, PortIn[kk___] :> {kk} ]; + in = FirstCase[{gg}, PortIn[kk___] :> {kk}]; in = qcPorts[-1, in, xx, yy]; - out = FirstCase[ {gg}, PortOut[kk___] :> {kk} ]; + out = FirstCase[{gg}, PortOut[kk___] :> {kk}]; out = qcPorts[+1, out, xx, yy]; Graphics[ Join[lines, in, nodes, marks, out], @@ -311,6 +311,10 @@ HoldPattern @ ] ] +(**** ****) + + +(**** *****) qcDepth::usage = "qcDepth[expr] returns the depth of quantum circuit element expr." @@ -318,9 +322,9 @@ qcDepth[qc_QuantumCircuit] := Total @ Map[qcDepth, List @@ qc] qcDepth[gg_List] := Max @ Map[qcDepth, gg] -qcDepth[any_] := Boole @ Not @ FreeQ[any, _?QubitQ | "Separator" | "Spacer" ] +qcDepth[any_] := Boole @ Not @ FreeQ[any, _?QubitQ | _?FermionQ | "Separator" | "Spacer" ] -(**** ****) +(**** *****) (**** *****) @@ -448,9 +452,6 @@ ParseGate[any_?StringQ, ___?OptionQ] := ( "Spacer" ) -ParseGate[expr_, ___?OptionQ] := expr /; FreeQ[expr, _?QubitQ] -(* Graphics primitives and directivescorrespond to this case. *) - (* Single-qubit gates *) @@ -467,7 +468,9 @@ ParseGate[ Map[ParseGate[#, opts]&, Reverse @ {ss}] -ParseGate[Measurement[ss:{___?PauliQ}, opts___?OptionQ], more___?OptionQ] := +ParseGate[Measurement[{}, ___?OptionQ], ___?OptionQ] = "Spacer" + +ParseGate[Measurement[ss:{__?SpeciesQ}, opts___?OptionQ], more___?OptionQ] := Map[ParseGate[Measurement[#], more, opts]&, ss] ParseGate[Measurement[S_?QubitQ, opts___?OptionQ], more___?OptionQ] := @@ -476,15 +479,19 @@ ParseGate[Measurement[S_?QubitQ, opts___?OptionQ], more___?OptionQ] := "Label" -> measurementLabel[S] ] +ParseGate[Measurement[c_?FermionQ, opts___?OptionQ], more___?OptionQ] := + Gate[{c}, "Shape" -> "Measurement", more, opts] + HoldPattern @ ParseGate[ Measurement[Multiply[ss__?QubitQ], opts___?OptionQ], more___?OptionQ ] := Gate[ FlavorMute @ {ss}, "Shape" -> "Measurement", more, opts, - "Label" -> measurementLabel@{ss} + "Label" -> measurementLabel @ {ss} ] + HoldPattern @ ParseGate[Projector[v_, qq_, ___?OptionQ], ___?OptionQ] := Gate[qq, "Shape" -> "Projector", "Label" -> None] @@ -708,6 +715,9 @@ ParseGate[Gate[args__, opts___?OptionQ], more___?OptionQ] := (* fallback *) +ParseGate[expr_, ___?OptionQ] := expr /; FreeQ[expr, _?QubitQ | _?FermionQ] +(* Graphics primitives and directivescorrespond to this case. *) + (* NOTE: This is dangerous because Plus and Times can happen. *) (* ParseGate[op_Symbol[any__, opts___?OptionQ], more__?OptionQ] := @@ -1103,7 +1113,7 @@ qcDrawGate::usage = "Renders the gates." qcDrawGate[gg_List, x_, yy_Association] := Map[qcDrawGate[#, x, yy]&, gg] qcDrawGate[ - Gate[cc:{Rule[_?QubitQ, _]..}, tt:{__?QubitQ}, opts___?OptionQ], + Gate[cc:{Rule[_?SpeciesQ, _]..}, tt:{__?SpeciesQ}, opts___?OptionQ], x_, yy_Association ] := Module[ @@ -1133,7 +1143,7 @@ qcDrawGate[ ] -qcDrawGate[Gate[tt:{__?QubitQ}, opts___?OptionQ], x_, yy_Association] := +qcDrawGate[Gate[tt:{__?SpeciesQ}, opts___?OptionQ], x_, yy_Association] := Module[ { yt = Lookup[yy, tt], new = FilterRules[Flatten@{opts}, Options @ Gate] }, @@ -1172,7 +1182,7 @@ qcLines[ gg_List, xx_List, yy_Association ] := Module[ Cases[ {#}, Gate[{S_?QubitQ}, "Shape" -> "Measurement", ___?OptionQ] -> S, Infinity ]&, gg - ]; + ]; mm = Flatten[ Thread /@ Thread[mm -> Most[xx]] ]; mm = KeySort @ KeyTake[Association @ mm, Keys @ yy]; diff --git a/Q3/Kernel/Quisso.wl b/Q3/Kernel/Quisso.wl index ca91c03a7..6deac430b 100644 --- a/Q3/Kernel/Quisso.wl +++ b/Q3/Kernel/Quisso.wl @@ -2413,26 +2413,32 @@ Projector[expr_, qq:{__?QubitQ}] := Measurement::usage = "Measurement[op] represents the measurement of Pauli operator op. Pauli operators include tensor products of the single-qubit Pauli operators.\nMeasurement[{op1, op2, \[Ellipsis]}] represents consecutive measurement of Pauli operators op1, op2, \[Ellipsis] in the reverse order." -Measurement::nonum = "Probability half is assumed for a state without explicitly numeric coefficients." - -Measurement::novec = "The expression `` does not seem to be a valid Ket expression. Null vector is returned." +Measurement::num = "Probability half is assumed for a state without explicitly numeric coefficients." SyntaxInformation[Measurement] = { "ArgumentsPattern" -> {_} } -Measurement[op_List][vec_] := - Construct[Composition @@ Map[Measurement, op], vec] +Measurement[op_?MatrixQ][in_?VectorQ] := theMeasurement[in, op] + +Measurement[op_List][in_] := Fold[theMeasurement, in , Reverse @ op] +(* NOTE: Here, Reverse is for backward compatibility. *) + +Measurement[op_][in_] := theMeasurement[in, op] + + +theMeasurement::usage = "The actual workhorse." -Measurement[op_][vec_] := Module[ - { odds = MeasurementOdds[op][vec], - rand = RandomReal[] }, - Garner @ If[ rand < Re @ First @ odds[0], +theMeasurement[in_, op_] := Module[ + { odds = MeasurementOdds[op][in], + rand = RandomReal[], + prob }, + prob = If[ NumericQ[First @ odds[0]], + Re[First @ odds[0]], + Message[Measurement::num]; + 0.5 + ]; + Garner @ If[ rand < prob, $MeasurementOut[op] = 0; Last @ odds[0], - $MeasurementOut[op] = 1; Last @ odds[1], - Message[Measurement::nonum]; - If[ rand < 1/2, - $MeasurementOut[op] = 0; Last @ odds[0], - $MeasurementOut[op] = 1; Last @ odds[1] - ] + $MeasurementOut[op] = 1; Last @ odds[1] ] ] @@ -2450,7 +2456,7 @@ Multiply[pre___, spr_Measurement, expr_] := Multiply[pre, spr @ expr] /; Not @ FreeQ[expr, _Ket] Measurement /: -Matrix[Measurement[op_], ss:{__?QubitQ}] := +Matrix[Measurement[op_], ss:{___?SpeciesQ}] := Measurement[Matrix[op, ss]] @@ -2459,29 +2465,24 @@ $MeasurementOut::usage = "$MeasurementOut gives the measurement results in an As Readout::usage = "Readout[expr, S] or Readout[expr, {S1, S2, ...}] reads the measurement result from the expr that is supposed to be the state vector after measurements." -Readout::nopauli = "`` is not a Pauli operator. Only Pauli operators (including tensor products of single-qubit Pauli operators) can be measured and readout." - Readout::notob = "`` (or some of its elements if it is a list) has never been measured. First use Measurement before using Readout." SyntaxInformation[Readout] = { "ArgumentsPattern" -> {_} } Readout[Measurement[op_]] := Readout[op] -Readout[op_] := ( - If[ Not @ KeyExistsQ[$MeasurementOut, op], - Message[Readout::notob, op] - ]; - $MeasurementOut[op] - ) -(* NOTE: op may be a matrix or an operator expression. *) +Readout[op_?MatrixQ] := $MeasurementOut[op] /; KeyExistsQ[$MeasurementOut, op] -Readout[op_List] := ( - If[ Not @ AllTrue[op, KeyExistsQ[$MeasurementOut, #]&], - Message[Readout::notob, op] - ]; - Lookup[$MeasurementOut, op] +Readout[op_List] := Lookup[$MeasurementOut, op] /; ContainsAll[Keys @ $MeasurementOut, op] +(* NOTE: op may be a list of matrices or operators. *) + +Readout[op_] := $MeasurementOut[op] /; KeyExistsQ[$MeasurementOut, op] + + +Readout[op_] := ( + Message[Readout::notob, op]; + $Failed ) -(* NOTE: op may be a list of matrices or operator expressions. *) MeasurementOdds::usage = "MeasurementOdds[op][vec] theoretically analyzes the process of meauring operator op on state vec and returns an association of elements of the form value->{probability, state}, where value is one of the possible measurement outcomes 0 and 1 (which correspond to eitemvalues +1 and -1, respectively, of op), probability is the probability for value to be actually observed, and state is the post-measurement state when value is actually observed." @@ -2490,7 +2491,8 @@ MeasurementOdds::pauli = "`` is not an observable Pauli operator." SyntaxInformation[MeasurementOdds] = { "ArgumentsPattern" -> {_} } -(* NOTE: DO NOT use op_?PauliQ; it will inerfere with mat_?MatrixQ below. *) +(* NOTE: DO NOT use op_?PauliQ; it will inerfere with mat_?MatrixQ below. + NOTE ADDED: Maybe not any longer (v3.4.4) because PauliQ and PauliMatrixQ are now separate. *) MeasurementOdds[op_][vec:(_State|_?fKetQ)] := Module[ { ss = Qubits[{vec, op}], aa, ff }, @@ -2520,8 +2522,8 @@ MeasurementOdds[mat_?MatrixQ][vec_?VectorQ] := Module[ 0 -> {p0/(p0+p1), pls}, 1 -> {p1/(p0+p1), mns} ] -] (* /; PauliQ[mat] *) -(* NOTE: Test PauliQ[mat] may take long for 8 or more qubits. *) +] (* /; PauliMatrixQ[mat] *) +(* NOTE: Test PauliMatrixQ[mat] may take long for 8 or more qubits. *) (* NOTE: 1. vec, pls, or mns may be 0 (null vector). 2. The norms of pls and mns may have imaginary parts numerically. diff --git a/Q3/Kernel/Wick.wl b/Q3/Kernel/Wick.wl index e5b7c0a08..2e2bc7c3c 100644 --- a/Q3/Kernel/Wick.wl +++ b/Q3/Kernel/Wick.wl @@ -2,16 +2,93 @@ BeginPackage["Q3`"] { WickLogarithmicNegativity, WickTimeReversalMoment }; -{ WickState, WickHistory, WickOperator, WickUnitary, - WickExpectation, WickGreensFunction }; +{ WickState, WickHistory, WickOperator, WickUnitary }; +{ WickExpectation, WickGreensFunction }; { WickCircuit, WickRandomCircuit }; { WickEntanglementEntropy }; { QuantumLog }; (* mainly in vonNeumann.wl *) +{ NambuMatrix, NambuRandomMatrix }; + Begin["`Private`"] +(**** ****) + +NambuMatrix::usage = "NambuMatrix[{a, b}, type] represents a 2n\[Times]2n matrix of type in the Nambu space, the n\[Times]n upper-left and upper-right blocks of which are given by a and b, respectively.\nNambuMatrix[mat] returns the pair {a, b}, where a and b are the n\[Times]n upper-left and upper-right blocks, respectively, of 2n\[Times]2n matrix mat.\nNambuMatrix[mat, type] returns NambuMatrix[{a, b}, type], where a and b are as above." + +NambuMatrix::odd = "Matrix `` has odd dimenions." + +NambuMatrix /: +MakeBoxes[op:NambuMatrix[trs:{_?MatrixQ, _?MatrixQ}, type_], fmt_] := + BoxForm`ArrangeSummaryBox[ + NambuMatrix, op, None, + { BoxForm`SummaryItem @ { "Type: ", type }, + BoxForm`SummaryItem @ { "Dimensions: ", Dimensions @ First @ trs } + }, + { BoxForm`SummaryItem @ { "Blocks: ", Map[MatrixObject, trs] } + }, + fmt, + "Interpretable" -> Automatic + ] /; ArrayQ[trs] + + +NambuMatrix[mat_?SquareMatrixQ] := Module[ + { n = (Length[mat] + 1) / 2 }, + Message[NambuMatrix::odd, mat]; + { mat[[ ;;n, ;;n]], Zero[n] } +] /; OddQ[Length @ mat] + +NambuMatrix[mat_?SquareMatrixQ] := With[ + { n = Length[mat] / 2 }, + { mat[[ ;;n, ;;n]], mat[[ ;;n, n+1;; ]] } +] + +NambuMatrix[mat_?SquareMatrixQ, type_] := + NambuMatrix[NambuMatrix[mat], type] + + +NambuMatrix /: + MatrixForm @ NambuMatrix[mm:{_, _}, _] := Map[MatrixForm, mm] + + +NambuMatrix /: + Normal @ NambuMatrix[{ham_, del_}, "Hermitian"] := ArrayFlatten @ { + {ham, del}, + {Topple[del], -Transpose[ham]} + } + +NambuMatrix /: + Normal @ NambuMatrix[{u_, v_}, "Unitary"] := ArrayFlatten @ { + {u, v}, + Conjugate @ {v, u} + } + +NambuMatrix /: + Normal @ NambuMatrix[{g_, f_}, "Green's"] := ArrayFlatten @ { + {g, f}, + {Topple[f], One[Dimensions @ g] - Transpose[g]} + } + + +NambuRandomMatrix::usage = "NambuRandomMatrix[n, type] returns a randomly generated matrix of type in the Nambu space with the n\[Times]n upper-left and upper-right blocks." + +NambuRandomMatrix[n_Integer, "Hermitian"] := + NambuMatrix[ + { RandomHermitian[n], + Symmetrize[RandomMatrix[n], Antisymmetric @ {1, 2}] }, + "Hermitian" + ] + +NambuRandomMatrix[n_Integer, "Unitary"] := With[ + { barH = NambuRandomMatrix[n, "Hermitian"] }, + NambuMatrix[MatrixExp[I * Normal[barH]], "Unitary"] +] + +(**** ****) + + (**** ****) (* See also: Shapourian and Ryu (2017, 2019) *) @@ -23,6 +100,9 @@ WickTimeReversalMoment::sing = "The matrix is tamed according to option \"Epsilo Options[WickTimeReversalMoment] = { "Epsilon" -> 1.25*^-5 } (* 1.0*^-8 is also a reasonable choice *) +WickTimeReversalMoment[alpha_, NambuMatrix[{g_?MatrixQ, f_?MatrixQ}, "Green's"], rest__] := + WickTimeReversalMoment[alpha, {g, f}, rest] + WickTimeReversalMoment[alpha_, grn_?MatrixQ, kk:{__Integer}, opts___?OptionQ] := WickTimeReversalMoment[alpha, {grn, Zero[Dimensions @ grn]}, kk, opts] @@ -30,7 +110,7 @@ WickTimeReversalMoment[alpha_, {grn_?MatrixQ, anm_?MatrixQ}, kk:{__Integer}, Opt { dd = Length[grn], gg, oo, id, xx, zz, uu, ww, pf, dgn, off }, - oo = Zero[dd, dd]; + oo = Zero[{dd, dd}]; id = One[dd]; xx = CircleTimes[ThePauli[1], id]; zz = CircleTimes[ThePauli[3], id]; @@ -42,10 +122,8 @@ WickTimeReversalMoment[alpha_, {grn_?MatrixQ, anm_?MatrixQ}, kk:{__Integer}, Opt Message[WickTimeReversalMoment::sing]; gg = tameMatrix[gg, OptionValue["Epsilon"]] ]; - gg = ArrayFlatten @ { - {gg, anm}, - {Topple @ anm, -Transpose[gg]} - }; + (* NOTE: When there is a fermion mode that is unoccuppied with certainty, the coherent-state representation becomes unusual, and one needs to handle such cases separately. While this is possible, Q3 offers a ditry trick. *) + gg = Normal @ NambuMatrix[{gg, -anm}, "Hermitian"]; pf = Power[Pfaffian[Inverse[gg.xx]], 2]; @@ -72,7 +150,7 @@ WickTimeReversalMoment[alpha_, {grn_?MatrixQ, anm_?MatrixQ}, kk:{__Integer}, Opt {ww[[dd+1;;, ;;dd]], oo} }; pf = Check[ - Pfaffian[(off - zz).xx] / pf, + Pfaffian[(off - zz).xx] / pf, Message[WickTimeReversalMoment::infy]; PseudoDivide[Pfaffian[(off - zz).xx], pf], Power::infy @@ -93,6 +171,7 @@ WickTimeReversalMoment[alpha_, {grn_?MatrixQ, anm_?MatrixQ}, kk:{__Integer}, Opt Total[Log[2, Power[gg, alpha] + Power[1-gg, alpha]]] + Log[2, Power[pf, alpha]] ] +tameMatrix::usage = "tameMatrix[mat, eps] tames a singular matrix mat with a small positive eps, and returns the resulting non-singular matrix." (* NOtE: mat is supposed to be Hermitian. *) tameMatrix[mat_?MatrixQ, eps_] := Module[ @@ -123,25 +202,31 @@ WickLogarithmicNegativity[grn_?MatrixQ, kk:{__Integer}, opts:OptionsPattern[]] : WickLogarithmicNegativity[{grn_?MatrixQ, anm_?MatrixQ}, kk:{__Integer}, opts:OptionsPattern[]] := WickTimeReversalMoment[1/2, {grn, anm}, kk, opts, "Epsilon" -> OptionValue["Epsilon"]] +WickLogarithmicNegativity[NambuMatrix[{g_?MatrixQ, f_?MatrixQ}, "Green's"], rest__] := + WickLogarithmicNegativity[{g, f}, rest] + (**** ****) -(**** ****) +(**** ****) -NambuMatrices /: -MakeBoxes[NambuMatrices[trs:{{_?MatrixQ, _?MatrixQ}..}], fmt_] := +BogoliubovTransforms /: +MakeBoxes[BogoliubovTransforms[trs:{{_?MatrixQ, _?MatrixQ}..}], fmt_] := BoxForm`ArrangeSummaryBox[ - "NambuMatrices", trs, None, - { BoxForm`SummaryItem @ { "Number of Transformations: ", Length @ trs }, - BoxForm`SummaryItem @ { "Dimensions of each: ", Dimensions @ trs[[1, 1]] } + "BogoliubovTransforms", trs, None, + { BoxForm`SummaryItem @ { "Transforms: ", Length @ trs }, + BoxForm`SummaryItem @ { "Dimensions: ", Dimensions @ trs[[1, 1]] } }, - { BoxForm`SummaryItem @ { "Transformations: ", Map[MatrixObject, trs, {2}] } + { BoxForm`SummaryItem @ { "Blocks: ", Map[MatrixObject, trs, {2}] } }, fmt, "Interpretable" -> Automatic ] /; ArrayQ[trs] -(**** ****) +(**** ****) + + +AddElaborationPatterns[_WickState, _WickUnitary, _WickOperator]; (**** ****) @@ -160,8 +245,8 @@ MakeBoxes[vec:WickState[pp:{___?FermionQ}, mm:{{_?MatrixQ, _?MatrixQ}...}, ff_?M BoxForm`SummaryItem @ { "Initial occupation: ", vec["Initial occupation"] } }, { BoxForm`SummaryItem @ { "Encoded modes: ", pp }, - BoxForm`SummaryItem @ { "Transformations: ", NambuMatrices @ mm }, - BoxForm`SummaryItem @ { "Wick matrix: ", ff } + BoxForm`SummaryItem @ { "Transformations: ", BogoliubovTransforms @ mm }, + BoxForm`SummaryItem @ { "Wick matrix: ", MatrixObject @ ff } }, fmt, "Interpretable" -> Automatic @@ -206,7 +291,37 @@ WickState[aa_Association] := WickState /: -Norm[ws_WickState] := Sqrt[Pfaffian[ws @ "Wick matrix"]] +Norm[ws_WickState] := Power[Det[ws @ "Wick matrix"], 1/4] +(* NOTE: The pfaffian of this Wick matrix must be positive. *) + + +WickState /: +Matrix[ws_WickState, rest___] := ws * Matrix[Ket[], rest] + +WickState /: +Matrix[ws:WickState[enc_List, trs:{{_?MatrixQ, _?MatrixQ}..}, mat_?MatrixQ, cc:{__?FermionQ}]] := Matrix[ws, cc] + +WickState /: +Matrix[ws:WickState[enc_List, trs:{{_?MatrixQ, _?MatrixQ}..}, mat_?MatrixQ, cc:{__?FermionQ}], ss:{__?SpeciesQ}] := + Module[ + { ff = Drop[enc, Length[enc]/2], + uu = Successive[theNambuDot[#2, theNambuTopple[#1]]&, trs] }, + ff = WickOperator /@ theWickDecode[cc] @ GroupBy[ff, First]; + uu = WickUnitary[#, cc]& /@ uu; + SparseArray @ Fold[ + Dot[Matrix[#2, ss], #1]&, + Matrix[Ket[cc], ss], + Riffle[Reverse @ Values @ ff, uu] + ] + ] + +WickState /: +Expand[ws:WickState[enc_List, trs:{{_?MatrixQ, _?MatrixQ}..}, mat_?MatrixQ, cc:{__?FermionQ}]] := + State[Matrix[ws, cc], cc] + +WickState /: +Elaborate[ws:WickState[enc_List, trs:{{_?MatrixQ, _?MatrixQ}..}, mat_?MatrixQ, cc:{__?FermionQ}]] := + ExpressionFor[Matrix[ws, cc], cc] (**** ****) @@ -232,21 +347,25 @@ WickHistory[ws_WickState, t_Integer] := Module[ WickHistory[{wc_WickCircuit, ws_WickState}] := Module[ { cc, id, new, n }, cc = Cases[First @ ws, c_[1, k_, 1] -> c[1, k, _], 1]; - id = WickUnitary @ theNambuOne[Length @ Last @ ws]; + id = WickUnitary[theNambuOne[Length @ Last @ ws], Last @ ws]; new = FoldList[wcEvent, {cc, {id}}, First @ wc]; wsClone[ws] /@ new ] +wsClone::usage = "wsClone[ws][{enc, trs}] returns a clone of Wick state ws by selecting encoded modes and blocks of the Wick matrix that are consistent with enc and replacing BdG transforms with trs." + wsClone[ws_WickState][{enc_, trs_}] := Module[ - { dd, mm, n }, + { dd, mm, wu, n }, dd = Cases[First @ ws, Alternatives @@ enc]; n = Length[dd] / 2; mm = wmTake[ws[[3]], n]; - WickState[dd, trs /. {WickUnitary -> Identity}, mm, Last @ ws] + WickState[dd, trs /. {wu_WickUnitary :> First[wu]}, mm, Last @ ws] ] +wcEvent::usage = "wcEvent[{enc, trs}, gate] updates {enc, trs} according to the new gate." + wcEvent[{enc_, trs_}, wu_WickUnitary] := Module[ { t = Max[First /@ enc], new }, @@ -269,15 +388,15 @@ wcEvent[{enc_, trs_}, Measurement[cc_]] := Module[ (**** ****) -(**** ****) +(**** ****) WickUnitary::usage = "WickUnitary[{u, v}] represents the unitary matrix ArrayFlatten[{{u, v}, Conjugate @ {v, u}}] in the Nambu space." WickUnitary /: -MakeBoxes[op:WickUnitary[trs:{_?MatrixQ, _?MatrixQ}], fmt_] := +MakeBoxes[op:WickUnitary[trs:{_?MatrixQ, _?MatrixQ}, ___?OptionQ], fmt_] := BoxForm`ArrangeSummaryBox[ WickUnitary, op, None, - { BoxForm`SummaryItem @ { "Type: ", "Nambu" }, + { BoxForm`SummaryItem @ { "Target: ", "Unspecified" }, BoxForm`SummaryItem @ { "Dimensions: ", Dimensions @ First @ trs } }, { BoxForm`SummaryItem @ { "Blocks: ", Map[MatrixObject, trs] } @@ -286,8 +405,35 @@ MakeBoxes[op:WickUnitary[trs:{_?MatrixQ, _?MatrixQ}], fmt_] := "Interpretable" -> Automatic ] /; ArrayQ[trs] +WickUnitary /: +MakeBoxes[op:WickUnitary[trs:{_?MatrixQ, _?MatrixQ}, cc:{___?FermionQ}, ___?OptionQ], fmt_] := + BoxForm`ArrangeSummaryBox[ + WickUnitary, op, None, + { BoxForm`SummaryItem @ { "Target: ", cc }, + BoxForm`SummaryItem @ { "Dimensions: ", Dimensions @ First @ trs } + }, + { BoxForm`SummaryItem @ { "Blocks: ", Map[MatrixObject, trs] } + }, + fmt, + "Interpretable" -> Automatic + ] /; ArrayQ[trs] + + +WickUnitary[NambuMatrix[{u_?MatrixQ, v_?MatrixQ}, "Unitary"], rest___] := + WickUnitary[{u, v}, rest] -WickUnitary[trs:{_?MatrixQ, _?MatrixQ}][ws_WickState] := +WickUnitary[mat_?SquareMatrixQ, rest___] := + WickUnitary[NambuMatrix @ mat, rest] + (* NOTE: Matrix mat is assumed to be unitary and have the block structure of a Bogoliubov-de Gennes transform. *) + +WickUnitary[1, cc:{___?FermionQ}, rest___] := + WickUnitary[theNambuOne[Length @ cc], cc, rest] + + +WickUnitary[trs:{_?MatrixQ, _?MatrixQ}, ___][fs_Ket] := + WickUnitary[trs][WickState @ fs] + +WickUnitary[trs:{_?MatrixQ, _?MatrixQ}, ___][ws_WickState] := Module[ { pp = ws["Encoded modes"], uu = ws["Transformations"], @@ -300,6 +446,100 @@ WickUnitary[trs:{_?MatrixQ, _?MatrixQ}][ws_WickState] := ReplacePart[ws, 2 -> uu] ] + +WickUnitary[trs:{_?MatrixQ, _?MatrixQ}, cc:{__?FermionQ}, ___][fs_Ket] := + WickUnitary[trs] @ WickState @ KetRegulate[fs, cc] + +WickUnitary[trs:{_?MatrixQ, _?MatrixQ}, cc:{__?FermionQ}, ___][ws_WickState] := + WickUnitary[trs][ws] /; cc == Last[ws] + + +WickUnitary /: +Times[z_, WickUnitary[uv_, rest___]] := WickUnitary[z * uv, rest] + +WickUnitary /: +Plus[WickUnitary[a_, cc_, opts___?OptionQ], WickUnitary[b_, cc_, more___?OptionQ]] := + WickUnitary[a + b, cc, Sequence @@ Flatten @ {opts, more}] + +WickUnitary /: +Plus[WickUnitary[a_, opts___?OptionQ], WickUnitary[b_, more___?OptionQ]] := + WickUnitary[a + b, Sequence @@ Flatten @ {opts, more}] + + +WickUnitary /: +Dot[WickUnitary[a_, cc_, opts___?OptionQ], WickUnitary[b_, cc_, more___?OptionQ]] := + WickUnitary[theNambuDot[a, b], cc, Sequence @@ Flatten @ {opts, more}] + +WickUnitary /: +Dot[WickUnitary[a_, opts___?OptionQ], WickUnitary[b_, more___?OptionQ]] := + WickUnitary[theNambuDot[a, b], Sequence @@ Flatten @ {opts, more}] + + +WickUnitary /: +Elaborate[wu:WickUnitary[{_?MatrixQ, _?MatrixQ}, cc:{__?FermionQ}, ___]] := + ExpressionFor[Matrix @ wu, cc] + +WickUnitary /: (* fallback *) +Elaborate[wu_WickUnitary] = wu + + +WickUnitary /: (* fallback *) +Matrix[wu_WickUnitary, rest___] := wu * Matrix[1, rest] + +WickUnitary /: +Matrix[wu:WickUnitary[{_?MatrixQ, _?MatrixQ}, cc:{__?FermionQ}, ___]] := + Matrix[wu, cc] + +WickUnitary /: +Matrix[WickUnitary[trs:{_?MatrixQ, _?MatrixQ}, cc:{__?FermionQ}, ___], ss:{__?SpeciesQ}] := Module[ + { ff = Select[ss, FermionQ], + rr = Select[ss, Not @* FermionQ], + mm, n }, + n = Length[ff]; + + (* Jordan-Wigner transformation *) + mm = Table[PadRight[Table[3, k-1], n], {k, n}] + 4 * One[n]; + mm = ThePauli /@ mm; + + ff = Thread[ff -> mm]; + rr = Thread[rr -> Map[One, Dimension @ rr]]; + + (* Convert the Nambu spinor to a list of matrices *) + mm = Values @ KeyTake[Join[ff, rr], cc]; + mm = Join[mm, Topple /@ mm]; + + MatrixExp @ TensorContract[ + Transpose[Topple /@ mm, {3, 1, 2}] . MatrixLog[Normal @ NambuMatrix[trs, "Unitary"]] . mm / 2, + {{2, 3}} + ] +] + + +WickUnitary /: +Normal @ WickUnitary[uv:{_?MatrixQ, _?MatrixQ}, ___] := + Normal @ NambuMatrix[uv, "Unitary"] + +WickUnitary /: +MatrixForm @ WickUnitary[uv:{_?MatrixQ, _?MatrixQ}, ___] := + Map[MatrixForm, uv] + +WickUnitary /: +NonCommutativeQ[_WickUnitary] = True + +WickUnitary /: +MultiplyKind[_WickUnitary] = Fermion + +WickUnitary /: +Multiply[pre___, wu_WickUnitary, ws_WickState] := Multiply[pre, wu @ ws] + +WickUnitary /: +Multiply[pre___, wu_WickUnitary, fs_Ket] := Multiply[pre, wu @ fs] + + +WickUnitary /: +ParseGate[WickUnitary[trs_, ff:{___?FermionQ}, opts___?OptionQ], more___?OptionQ] := + Gate[ff, more, opts, "Label" -> "U"] + (**** ****) @@ -315,17 +555,21 @@ WickOperator[trs:{_?MatrixQ, _?MatrixQ}] := WickUnitary[trs] +WickOperator[op:{___?AnyFermionQ}][fs_Ket] := + WickOperator[op][WickState @ fs] + +WickOperator[{}][ws_WickState] = ws + + (* For the vacuum state *) -WickOperator[op:{__?AnyFermionQ}][WickState[{}, {uv_}, {{}}, cc:{__?FermionQ}]] := +WickOperator[op:{__?AnyFermionQ}, ___][WickState[{}, {uv_}, {{}}, cc:{__?FermionQ}]] := Module[ {enc, trs, mat}, {enc, trs, mat} = wmBuild[cc][op, uv]; WickState[enc, trs, mat, cc] ] -WickOperator[{}][ws_WickState] = ws - -WickOperator[op:{__?AnyFermionQ}][ws_WickState] := +WickOperator[op:{__?AnyFermionQ}, ___][ws_WickState] := Module[ { enc, trs, mat, dd }, {enc, trs, mat} = List @@ Most[ws]; @@ -335,12 +579,47 @@ WickOperator[op:{__?AnyFermionQ}][ws_WickState] := WickState[enc, trs, mat, Last @ ws] ] + +WickOperator /: +NonCommutativeQ[_WickOperator] = True + +WickOperator /: +MultiplyKind[_WickOperator] = Fermion + +WickOperator /: +Multiply[pre___, opr_WickOperator, ws_WickState] := Multiply[pre, opr[ws]] + +WickOperator /: +Multiply[pre___, opr_WickOperator, fs_Ket] := Multiply[pre, opr[WickState @ fs]] + + +WickOperator /: +Matrix[WickOperator[cc:{__?AnyFermionQ}, ___], rest___] := + Dot @@ Matrix[cc, rest] + +WickOperator /: (* fallback *) +Matrix[op_WickOperator, rest___] := op * Matrix[1, rest] + +WickOperator /: +Elaborate[op:WickOperator[{__?AnyFermionQ}, ___]] := + ExpressionFor[Matrix[op], Fermions @ op] + +WickOperator /: (* fallback *) +Elaborate[op_WickOperator] = op + + +WickOperator /: +ParseGate[WickOperator[op:{___?AnyFermionQ}, opts___?OptionQ], more___?OptionQ] := + Gate[Fermions @ op, more, opts] + (**** ****) (**** ****) -Measurement[c_?FermionQ][ws_WickState] := Module[ +theMeasurement[fs_Ket, c_?FermionQ] := theMeasurement[WickState @ fs, c] + +theMeasurement[ws_WickState, c_?FermionQ] := Module[ { tt = ws["Stages"], cc = ws["Bare modes"], aa = ws["Encoded modes"], @@ -356,7 +635,8 @@ Measurement[c_?FermionQ][ws_WickState] := Module[ (* Simulate the measurement process. *) (* NOTE: WickState is unnormalized. *) - If[ RandomReal[] < Chop[Pfaffian @ nn / Pfaffian @ mm], + (* NOTE: Both nn and mm have a positive Pfaffian. *) + If[ RandomReal[] < Chop[Sqrt[Det @ nn / Det @ mm]], $MeasurementOut[c] = 0, $MeasurementOut[c] = 1; op = Reverse @ op; (* op := Dagger[c] ** c *) @@ -370,6 +650,13 @@ Measurement[c_?FermionQ][ws_WickState] := Module[ WickState[bb, vv, nn, cc] ] + +Measurement /: +Multiply[msr:Measurement[_?FermionQ|{__?FermionQ}], ws_WickState] := msr[ws] + +Measurement /: +Multiply[msr:Measurement[_?FermionQ|{__?FermionQ}], fs_Ket] := msr[WickState @ fs] + (**** ****) @@ -403,6 +690,9 @@ WickExpectation[ws_WickState][expr_Plus] := WickExpectation[ws_WickState][HoldPattern @ Multiply[ops__?AnyFermionQ]] := WickExpectation[ws][{ops}] +WickExpectation[ws_WickState][WickOperator[ops:{__?AnyFermionQ}, ___]] := + WickExpectation[ws][ops] + WickExpectation[ws_WickState][ops:{__?AnyFermionQ}] := Module[ { tt = ws["Stages"], cc, dd, pp, uv, mm, gg }, @@ -424,39 +714,44 @@ WickGreensFunction[ws_WickState] := WickGreensFunction[ws_WickState, dd:{___?FermionQ}] := Module[ { tt = ws["Stages"], - aa, bb, pp, uv, mm, ff, gg, L, n, i, j }, + aa, bb, pp, uv, mm, ff, gg, i, j, k, n }, {pp, uv, mm, cc} = List @@ ws; n = Length[dd]; - L = Length[pp] / 2; + k = Length[pp] / 2; (* encoded modes *) aa = theWickCode[cc][dd, {tt}]; bb = dagWickCode /@ aa; (* NOT dagWickCode[aa] *) (* Green's function *) - gg = Table[ - {i, i} -> Re @ Pfaffian @ Last @ wmWedge[{aa[[i]], bb[[i]]}, pp, uv, mm, L], + gg = Zero[{n, n}]; + Table[ + gg[[i, i]] = Re @ Sqrt @ Det @ Last @ wmWedge[{aa[[i]], bb[[i]]}, pp, uv, mm, k], {i, 1, n} - ]; (* supposed to be real *) - gg = Join[ gg, - Flatten @ Table[ - {i, j} -> Pfaffian @ Last @ wmWedge[{aa[[i]], bb[[j]]}, pp, uv, mm, L], - {i, 1, n}, - {j, i+1, n} - ] + ]; (* NOTE: The Pfaffians here are supposed to be real positive. *) + Table[ + gg[[i, j]] = Pfaffian @ Last @ wmWedge[{aa[[i]], bb[[j]]}, pp, uv, mm, k]; + gg[[j, i]] = Conjugate @ gg[[i, j]], + {i, 1, n}, + {j, i+1, n} ]; - gg = SymmetrizedArray[gg, {n, n}, Hermitian @ {1, 2}]; + (* gg = SymmetrizedArray[gg, {n, n}, Hermitian @ {1, 2}]; *) + (* NOTE: Too dense and pointless for most purposes. *) (* anomalous Green's function *) - ff = Flatten @ Table[ - {i, j} -> Pfaffian @ Last @ wmWedge[{aa[[i]], aa[[j]]}, pp, uv, mm, L], + ff = Zero[{n, n}]; + Table[ + ff[[i, j]] = Pfaffian @ Last @ wmWedge[{aa[[i]], aa[[j]]}, pp, uv, mm, k]; + ff[[j, i]] = -ff[[i, j]], {i, 1, n}, {j, i+1, n} ]; - ff = SymmetrizedArray[ff, {n, n}, Antisymmetric @ {1, 2}]; + (* ff = SymmetrizedArray[ff, {n, n}, Antisymmetric @ {1, 2}]; *) + (* NOTE: Too dense and pointless for most purposes. *) - {gg, ff} / Re[Pfaffian @ mm] (* This Pfaffian is supposed to be real. *) + NambuMatrix[{gg, ff} / Re[Sqrt @ Det @ mm], "Green's"] + (* NOTE: The Pfaffian of mm is supposed to be real positive. *) ] (**** ****) @@ -478,16 +773,15 @@ WickLogarithmicNegativity[ws_WickState, dd:{__?FermionQ}, opts:OptionsPattern[]] (**** ****) -(**** ****) +(**** ****) (* See, e.g., Calabrese and Carday (2004) and Peschel (2003). *) WickEntanglementEntropy::usage = "WickEntanglementEntropy[ws, {d1, d2, \[Ellipsis]}] returns the entanglement entropy in the Wick state ws between the subsystem {d1, d2, \[Ellipsis]} \[Subset] {c1, c2, \[Ellipsis], cn} and the rest.\nWickEntanglementEntropy[{grn, anm}, {k1, k2, \[Ellipsis]}] calculates the entanglement entropy from the normal (grn) and anomalous (anm) Green's functions. The index set {k1, k2, \[Ellipsis]} specifies the subsystem." WickEntanglementEntropy[ws_WickState, dd:{__?FermionQ}] := Module[ - { gg = WickGreensFunction[ws, dd], + { gg = Normal @ WickGreensFunction[ws, dd], id, ff }, - gg = theNambuGreen[gg]; id = One[Dimensions @ gg]; (QuantumLog[gg] + QuantumLog[id - gg]) / 2 ] @@ -496,14 +790,17 @@ WickEntanglementEntropy[{grn_?MatrixQ, anm_?MatrixQ}, kk:{__Integer}] := Module[ { gg = Normal[grn][[kk, kk]], ff = Normal[anm][[kk, kk]], id }, - gg = theNambuGreen[{gg, ff}]; + gg = Normal @ NambuMatrix[{gg, ff}, "Green's"]; (* gg = SymmetrizedArray[gg, Dimensions @ gg, Hermitian @ {1,2}]; *) id = One[Dimensions @ gg]; (QuantumLog[gg] + QuantumLog[id - gg]) / 2 ] (* NOTE: It seems that Part does not support properly SymmetrizedArray, hence Normal is required in the above. *) -(**** ****) +WickEntanglementEntropy[NambuMatrix[mm:{_?MatrixQ, _?MatrixQ}, "Green's"], kk_] := + WickEntanglementEntropy[mm, kk] + +(**** ****) (**** ****) @@ -543,13 +840,16 @@ wmBuild[cc:{__?FermionQ}][spec:({___?AnyFermionQ} | {_?MatrixQ, _?MatrixQ})...] enc = Join[dagWickCode @ enc, enc]; n = Length[enc]; - mat = Flatten @ Table[ - {i, j} -> theWickContract[trs][enc[[i]], enc[[j]]], + mat = Zero[{n, n}]; + Table[ + mat[[i, j]] = theWickContract[trs][enc[[i]], enc[[j]]]; + mat[[j, i]] = -mat[[i, j]], {i, n}, {j, i+1, n} ]; - mat = SymmetrizedArray[mat, {n, n}, Antisymmetric[{1, 2}]]; - {enc, trs, mat} + (* mat = SymmetrizedArray[mat, {n, n}, Antisymmetric[{1, 2}]]; *) + (* NOTE: Too dense and pointless for most purposes. *) + {enc, trs, Normal @ mat} ] (**** ****) @@ -557,7 +857,7 @@ wmBuild[cc:{__?FermionQ}][spec:({___?AnyFermionQ} | {_?MatrixQ, _?MatrixQ})...] (**** ****) -wmWedge::usage = "wmWedge[dd, pp, mm, mat, n] cleaves the Wick matrix mat at row and column n and wedge into there additional blocks with rows and columns corresponding to the new fermion modes dd." +wmWedge::usage = "wmWedge[dd, pp, mm, mat, k] cleaves the Wick matrix mat at row and column k and wedge into there additional blocks with rows and columns corresponding to the new fermion modes dd." (* INPUT dd : the list of new fermion modes to wedge into the Wick matrix mat. @@ -566,47 +866,45 @@ wmWedge::usage = "wmWedge[dd, pp, mm, mat, n] cleaves the Wick matrix mat at row mat: the Wick matrix to updated n : the row and column where the new modes to be wedged. *) + (* NOTE: Both dd and pp are given in the form of encoded modes. *) wmWedge[ dd:{__?FermionQ}, (* encoded modes *) - pp:{__?FermionQ}, uv:{{_?MatrixQ, _?MatrixQ}...}, mm_SymmetrizedArray, - n_Integer] := Module[ - { m = Length[dd], - p1, p2, rr, ss, i, j }, - - {p1, p2} = TakeDrop[pp, n]; - - (* crack open the matrix. *) - rr = Most[SymmetrizedArrayRules @ mm] /. { - {i_, j_} /; i > n && j > n -> {i + m, j + m}, - {i_, j_} /; i > n -> {i + m, j}, - {i_, j_} /; j > n -> {i, j + m} - }; + pp:{__?FermionQ}, uv:{{_?MatrixQ, _?MatrixQ}...}, mat_?MatrixQ, + k_Integer] := Module[ + { new = mat, + p1, p2, rr, ss, i, j, n }, + n = Length[dd]; + {p1, p2} = TakeDrop[pp, k]; + + (* Crack open the matrix. *) + new = Zero[Dimensions[mat] + n]; + new[[ ;;k, ;;k]] = mat[[ ;;k, ;;k]]; + new[[ ;;k, n+k+1;; ]] = mat[[ ;;k, k+1;; ]]; + new[[n+k+1;; , ;;k]] = mat[[k+1;; , ;;k]]; + new[[n+k+1;; , n+k+1;; ]] = mat[[k+1;; , k+1;; ]]; (* wedge additional blocks into the crack. *) - ss = Flatten @ Table[ - {i, j+n} -> theWickContract[uv][p1[[i]], dd[[j]]], - {i, n}, - {j, m} + Table[ + new[[i, j+k]] = theWickContract[uv][p1[[i]], dd[[j]]]; + new[[j+k, i]] = -new[[i, j+k]], + {i, k}, + {j, n} ]; - rr = Join[rr, ss]; - ss = Flatten @ Table[ - {i+n, j+n} -> theWickContract[uv][dd[[i]], dd[[j]]], - {i, m}, - {j, i+1, m} + Table[ + new[[i+k, j+k]] = theWickContract[uv][dd[[i]], dd[[j]]]; + new[[j+k, i+k]] = -new[[i+k, j+k]], + {i, n}, + {j, i+1, n} ]; - rr = Join[rr, ss]; - ss = Flatten @ Table[ - {i+n, j+n+m} -> theWickContract[uv][dd[[i]], p2[[j]]], - {i, m}, - {j, Length[pp] - n} + Table[ + new[[i+k, j+k+n]] = theWickContract[uv][dd[[i]], p2[[j]]]; + new[[j+k+n, i+k]] = -new[[i+k, j+k+n]], + {i, n}, + {j, Length[pp] - k} ]; - rr = Join[rr, ss]; (* Reconstruct the Wick matrix. *) - { Join[p1, dd, p2], - uv, - SymmetrizedArray[rr, Dimensions[mm] + {m, m}, Antisymmetric[{1, 2}]] - } + { Join[p1, dd, p2], uv, Normal @ new } ] (**** ****) @@ -616,20 +914,10 @@ wmWedge[ dd:{__?FermionQ}, (* encoded modes *) wmTake::usage = "wmTake[mat,k] takes the first and last k rows and columns of Wick matrix mat." -wmTake[mat_SymmetrizedArray, k_Integer] := Module[ - { n = First[Dimensions@mat], - rr }, - rr = Most@SymmetrizedArrayRules[mat]; - rr = Join[ - Normal @ KeySelect[rr, First[#] <= k && Last[#] <= k &], - Normal @ KeySelect[rr, First[#] <= k && Last[#] > n-k &], - Normal @ KeySelect[rr, First[#] > n-k && Last[#] > n-k &] - ]; - rr = KeyReplace[ rr, - { {i_, j_} /; i > n-k && j > n-k -> {i - n + 2*k, j - n + 2*k}, - {i_, j_} /; j > n-k -> {i, j - n + 2*k} } - ]; - SymmetrizedArray[rr, {2 k, 2 k}, Antisymmetric@{1, 2}] +wmTake[mat_?MatrixQ, k_Integer] := Module[ + { rr }, + rr = List /@ Range[1+k, Length[mat]-k]; + Delete[Transpose @ Delete[Transpose @ mat, rr], rr] ] (**** ****) @@ -684,6 +972,15 @@ dagWickCode[ff:{f_[_, _, _]...}] := Reverse @ Map[dagWickCode, ff] dagWickCode[f_[t_, k_, d_]] := f[t, k, Mod[d+1, 2]] + +theWickDecode::usage = "theWickDecode[cc][c[t_, k_, type_]] returns cc[[k]] if type is 0 and Dagger[cc[[kk]]] if type is 1." + +theWickDecode[cc:{__?FermionQ}][ff:(_List|_Association)] := Map[theWickDecode[cc], ff] + +theWickDecode[cc:{__?FermionQ}][_[t_, k_, 0]] := cc[[k]] + +theWickDecode[cc:{__?FermionQ}][_[t_, k_, 1]] := Dagger[ cc[[k]] ] + (**** ****) @@ -699,29 +996,40 @@ theNambuDot[{u1_?MatrixQ, v1_?MatrixQ}, {u2_?MatrixQ, v2_?MatrixQ}] := {u1.u2 + v1.Conjugate[v2], u1.v2 + v1.Conjugate[u2]} (* Convention: barc_i --> barU_{ij}barc_j with barU := {{U, V}, Conjugate @ {V, U}} *) + +theNambuDot[a:WickUnitary[_, cc_, ___], b:WickUnitary[_, cc_, ___]] := + WickUnitary[theNambuDot[First @ a, First @ b], cc, Join[Options @ a, Options @ b]] + theNambuDot[a_WickUnitary, b_WickUnitary] := WickUnitary @ theNambuDot[First @ a, First @ b] + +theNambuDot[a:NambuMatrix[_, "Unitary"], b:NambuMatrix[_, "Unitary"]] := + NambuMatrix[theNambuDot[First @ a, First @ b], "Unitary"] + (**** ****) -theNambuOne::usage = "theNambuOne[n] returns the pair {One[n], Zero[n,n]}." +(**** ****) -theNambuOne[n_Integer] := {One @ n, Zero @ {n, n}} +theNambuTopple[{u_?MatrixQ, v_?MatrixQ}] := {Topple @ u, Transpose @ v} +theNambuTopple[wu_WickUnitary] := WickUnitary[First @ wu, Sequence @@ Flatten[Options @ wu]] -toNambuForm::usage = "toNambuForm[mat] ..." +theNambuTopple[NambuMatrix[uv:{_?MatrixQ, _?MatrixQ}, "Unitary"]] := + NambuMatrix[First @ uv, "Unitary"] -toNambuForm[m_?MatrixQ] := {m, Zero @ Dimensions @ m} +(**** ****) -theNambuGreen::usage = "theNambuGreen[{g, f}] ..." +theNambuOne::usage = "theNambuOne[n] returns the pair {One[n], Zero[{n, n}]}." -theNambuGreen[{g_?MatrixQ, f_?MatrixQ}] := ArrayFlatten @ { - {g, f}, - {Topple[f], One[Dimensions @ g] - Transpose[g]} -} +theNambuOne[n_Integer] := {One @ n, Zero @ {n, n}} + + +toNambuForm::usage = "toNambuForm[mat] ..." +toNambuForm[m_?MatrixQ] := {m, Zero @ Dimensions @ m} (**** ****) @@ -735,22 +1043,28 @@ Options[WickRandomCircuit] = { "SaveData" -> False, "Overwrite" -> True, "Filename" -> Automatic, - "Data" -> "States", + "Output" -> "States", "Prefix" -> "WRC" } -WickRandomCircuit[cc_, uv:{_?MatrixQ, _?MatrixQ}, rest___] := - WickRandomCircuit[cc, WickUnitary @ uv, rest] +WickRandomCircuit[cc_, trs:{_?MatrixQ, _?MatrixQ}, rest___] := + WickRandomCircuit[cc, WickUnitary[trs, cc], rest] + +WickRandomCircuit[cc_, NambuMatrix[trs:{_?MatrixQ, _?MatrixQ}, "Unitary"], rest___] := + WickRandomCircuit[cc, WickUnitary[trs, cc], rest] + +WickRandomCircuit[cc_, WickUnitary[trs:{_?MatrixQ, _?MatrixQ}, opts___?OptionQ], rest___] := + WickRandomCircuit[cc, WickUnitary[trs, cc, opts], rest] WickRandomCircuit[ cc:{__?FermionQ}, - wu:WickUnitary[{_?MatrixQ, _?MatrixQ}], + wu:WickUnitary[{_?MatrixQ, _?MatrixQ}, {__?FermionQ}, ___], p_?NumericQ, t_Integer, OptionsPattern[]] := Module[ { k = 0, progress = 0, program, data, - in, qc, n, m }, + in, qc, qc2, n, m }, PrintTemporary @ ProgressIndicator @ Dynamic[progress]; (* initial state *) @@ -771,16 +1085,16 @@ Module[ m ] }; - Switch[ OptionValue["Data"], + Switch[ OptionValue["Output"], "States", Null, "Correlations", program := Table[ progress = ++k / (n*m); - FoldPairList[{WickGreensFunction[#1], Construct[#2, #1]}&, in, qc], + FoldPairList[{WickGreensFunction[#1], Construct[#2, #1]}&, in, Append[qc, WickUnitary[1, cc]]], m ], - _, Message[WickRandomCircuit::data, OptionValue["Data"]] + _, Message[WickRandomCircuit::data, OptionValue["Output"]] ]; data = Table[ (* quantum circuit with randomly selected measurements *) @@ -788,7 +1102,7 @@ Module[ program, n ]; - Switch[ OptionValue["Data"], + Switch[ OptionValue["Output"], "Correlations", data = Flatten[data, 1] ]; @@ -809,7 +1123,7 @@ Module[ If[OptionValue["Overwrite"] && FileExistsQ[file], DeleteFile @ file]; - Export[file, Association @ {"Times" -> tt, "Data" -> data}] + Export[file, data] ] (**** ****) @@ -832,6 +1146,14 @@ MakeBoxes[wc:WickCircuit[gg:{(_Measurement | _WickUnitary)..}, cc:{___?FermionQ} "Interpretable" -> Automatic ] +WickCircuit /: + Graphics[wc_WickCircuit, opts___?OptionQ] := + Graphics[QuantumCircuit @@ First[wc], opts] + +WickCircuit /: + Show[wc_WickCircuit, opts___?OptionQ] := + Graphics[QuantumCircuit @@ First[wc], opts] + (**** ****) diff --git a/Q3/Kernel/Wimmer.wl b/Q3/Kernel/Wimmer.wl index 42f051e48..d747a380a 100644 --- a/Q3/Kernel/Wimmer.wl +++ b/Q3/Kernel/Wimmer.wl @@ -5,68 +5,112 @@ BeginPackage["Q3`"] { Pfaffian, SkewTridiagonalize }; +{ Zero, One }; (* mainly in Pauli *) + +{ PfaffianMine }; + Begin["`Private`"] +(**** ****) +(**** REMARK: + Depending on the type of entries, different methods must chosen. However, for large arrays, the type test itself takes time. Therefore, it is important to have an efficient test method. + *) + +arrayRealQ::usage = "\ +arrayRealQ[array] returns True if array is an array of real numbers.\n\ +arrayRealQ[array, test] first invokes ArrayQ[array, 1|2, test]." + +arrayRealQ[obj_, test_] := And[ + ArrayQ[obj, 1|2, test], + arrayRealQ[obj] +] + +arrayRealQ[obj_SymmetrizedArray] := + FreeQ[Values @ SymmetrizedArrayRules @ obj, _Complex] + +arrayRealQ[obj_SparseArray] := + FreeQ[obj["ExplicitValues"], _Complex] + +arrayRealQ[obj_] := + FreeQ[obj, _Complex] +(* NOTE: FreeQ is much fastern than, e.g., MatrixQ[obj, Not[MatchQ[#, _Complex]]&] *) + +(**** ****) + + (**** ****) Pfaffian::usage = "Pfaffian[mat] returns the Pfaffian of an anti-symmetric matrix mat." -Pfaffian::number = "Non-numerical matrix `` is given; Pfaffian can be computed efficiently only for a numerical matrix. If the matrix is small enough, try option Method -> \"Heuristic\"." +Pfaffian::method = "Unrecognized option `1`; must be Automatic, \"Parlett-Reid\", \"Householder\", \"Hessenberg\" or \"Heuristic\"."; -Pfaffian::method = "Unrecognized option `1`; must be either \"ParlettReid\", \"Householder\", \"Hessenberg\" or \"Heuristic\"."; +Pfaffian::number = "Non-numerical matrix `` is given; Pfaffian can be computed efficiently only for a numerical matrix. If the matrix is small enough, try option Method -> \"Heuristic\"." -Pfaffian::Hessenberg = "Pfaffian computation with Hessenberg decomposition only works for real matrices; instead, using Householder method."; +Pfaffian::Hessenberg = "Pfaffian computation with Hessenberg decomposition only works for real matrices of finite precision; using the Parlett-Reid method instead."; +Pfaffian::sparse = "Too density for a sparse array; converted to a normal form." Options[Pfaffian] = {Method -> Automatic} -Pfaffian[mat_?SquareMatrixQ] := 0 /; OddQ[Length @ mat] + +Pfaffian[mat_SymmetrizedArray, opts:OptionsPattern[]] := + Pfaffian[SparseArray @ mat, opts] +(* NOTE: The current implementation of Pfaffian breaks the structured array. *) + +Pfaffian[mat_SparseArray, opts:OptionsPattern[]] := + ( Message[Pfaffian::sparse]; + Pfaffian[Normal @ mat, opts] + ) /; mat["Density"] > 0.2 + + +Pfaffian[mat_?SquareMatrixQ, OptionsPattern[]] := 0 /; OddQ[Length @ mat] Pfaffian[mat_?SquareMatrixQ, OptionsPattern[]] := + PfaffianHeuristic[mat] /; OptionValue[Method] == "Heuristic" + + +Pfaffian[mat_, OptionsPattern[]] := Switch[ OptionValue[Method], Automatic, - Which[ - (* real and finite precision *) - MatrixQ[mat, NumericQ[#] && Not[MatchQ[#, _Complex]] &], - PfaffianHessenberg[mat], - (* real but possibly infinite precision *) - MatrixQ[mat, NumerQ[#] && Not[MatchQ[#, _Complex]] &], - PfaffianHouseholderReal[mat], - True, - PfaffianHouseholderComplex[mat] + If[ arrayRealQ[mat], + PfaffianHessenberg[N @ mat], + PfaffianLTL[mat] ], - "ParlettReid", PfaffianLTL[mat], + "Parlett-Reid", PfaffianLTL[mat], "Householder", PfaffianHouseholder[mat], "Hessenberg", PfaffianHessenberg[mat], - "Heuristic", PfaffianHeuristic[mat], _, Message[Pfaffian::method, OptionValue @ Method]; 0 - ] /; MatrixQ[mat, NumberQ] - -Pfaffian[mat_?SquareMatrixQ, OptionsPattern[]] := - (Message[Pfaffian::number, mat]; Indeterminate) + ] /; MatrixQ[mat, NumericQ] (**** ****) (**** ****) -SkewTridiagonalize::usage = "SkewTridiagonalize[mat] tridiagonalizes the skew-symmetric matrix mat and returns {T, L, P} such that P.mat.P^T = L.mat.L^T, where T is a skew-symmetric tridiagonal matrix.\nIn the Parlett-Reid method, L is a unit lower triangular matrix and P is a permutation matrix.\nIn the Householder method, L is a unitary matrix and P is the identity matrix." +SkewTridiagonalize::usage = "SkewTridiagonalize[mat] tridiagonalizes the skew-symmetric matrix mat and returns {T, L, P} such that P.mat.P^T = L.T.L^T, where T is a skew-symmetric tridiagonal matrix.\nIn the Parlett-Reid method, L is a unit lower triangular matrix and P is a permutation matrix.\nIn the Householder method, L is a unitary matrix and P is the identity matrix." -SkewTridiagonalize::method = "Unrecognized option `1`; must be either \"ParlettReid\", \"Householder\" or \"Hessenberg\"."; +SkewTridiagonalize::method = "Unrecognized option `1`; must be Automatic, \"Parlett-Reid\", \"Householder\", \"Hessenberg\" or \"Heuristic\"."; + +SkewTridiagonalize::number = "Non-numerical matrix `` is given; the tridiagonalization can be computed efficiently only for a numerical matrix." SkewTridiagonalize::Hessenberg = "SkewTridiagonalize with the Hessenberg decomposition only works for real matrices; instead, using Householder method."; -Options[SkewTridiagonalize] = {Method -> "ParlettReid"} +Options[SkewTridiagonalize] = {Method -> Automatic} -SkewTridiagonalize[mat_?SquareMatrixQ, OptionsPattern[]] := +SkewTridiagonalize[mat_, OptionsPattern[]] := Switch[ OptionValue[Method], - "ParlettReid", SkewLTL[mat], + Automatic, + Which[ + arrayRealQ[mat], SkewHessenberg[N @ mat], (* cf. SkewHouseholderReal *) + True, SkewLTL[mat] + ], + "Parlett-Reid", SkewLTL[mat], "Householder", SkewHouseholder[mat], "Hessenberg", SkewHessenberg[mat], - _, Message[SkewTridiagonalize::method, OptionValue @ Method]; 0 - ] + _, Message[Skew::method, OptionValue @ Method]; 0 + ] /; MatrixQ[mat, NumericQ] (**** ****) @@ -76,9 +120,8 @@ SkewTridiagonalize[mat_?SquareMatrixQ, OptionsPattern[]] := (* PosiitionLargest used in PfaffianLTL and SkewLTL was introduced Mathematica v13.2. *) If[ $VersionNumber < 13.2, - PositionLargest[list_?VectorQ] := - Flatten @ Position[Normal @ list, Max @ list] - (* NOTE: Normal is required here for older version of Mathematica. *) + PositionLargest[list_?VectorQ] := FirstPosition[Normal @ list, Max @ list] + (* NOTE: Normal is required here for older versions of Mathematica. *) ]; (**** ****) @@ -86,6 +129,37 @@ If[ $VersionNumber < 13.2, (**** ****) +PfaffianLTL::usage = "PfaffianLTL[mat] computes the Pfaffian of the skew-symmetric matirx mat using the L T L^T decomposition." + +PfaffianLTL[mat_] := Module[ + { new = mat, + val = 1, + pos, aa, bb }, + If[OddQ[Length @ new], Return[0]]; + While[ Length[new] > 2, + pos = First @ PositionLargest[Abs @ First @ new]; + If[ZeroQ @ new[[1, pos]], Return[0]]; + If[ pos != 2, + new[[{2, pos}, ;;]] = new[[{pos, 2}, ;;]]; + new[[;;, {2, pos}]] = new[[;;, {pos, 2}]]; + val *= -1 + ]; + val *= new[[1, 2]]; + + aa = new[[1, 3;;]] / new[[1, 2]]; + bb = new[[2, 3;;]]; + new = new[[3;;, 3;;]]; + new -= (# - Transpose[#]&) @ Outer[Times, aa, bb]; + ]; + Return[val * new[[1, 2]]] +] /; MatrixQ[mat, NumericQ] + +PfaffianLTL[mat_] := ( + Message[Pfaffian::number, mat]; + Return[Indeterminate] +) + + SkewLTL::usage = "SkeyLTL[mat] computes the L T L^T decomposition of skew-symmetric matrix mat using the Parlett-Reid algorithm, and returns T, L and P, where T is a tridiagonal matrix, L a unit lower triangular matrix and P a permutation matrix such that P A P^T=L T L^T." (* NOTE: This function is not needed for the Pfaffian computation, but is only provided for demonstration purposes. *) @@ -120,46 +194,6 @@ SkewLTL[Mat_] := Module[ Return[{A, L, SparseArray[{i_, i_} -> 1, {N, N}][[Pv]]}] ] - -PfaffianLTL::usage = "PfaffianLTL[mat] computes the Pfaffian of the skew-symmetric matirx mat using the L T L^T decomposition." - -PfaffianLTL[mat_SymmetrizedArray] := PfaffianLTL[SparseArray @ mat] -(* NOTE: PfaffianLTL involves interchange of certain pairs of rows and columns, which breaks the symmetry in mat and causes error. *) - -PfaffianLTL[mat_] := Module[ - { new = mat, - dim = Length @ mat, - val, ip}, - If[OddQ[dim], Return[0]]; - val = 1; - For[i = 1, i < dim-1, i += 2, - (* Find the maximum entry in column i, starting from row i+1. *) - ip = i + First @ PositionLargest[Abs @ new[[i+1;;, i]]]; - (* If the maximum entry is not at i+1, permute the matrix so that it is. *) - If[ i+1 != ip, - (* Interchange rows and columns in new. *) - new[[{i+1, ip}, ;;]] = new[[{ip, i+1}, ;;]]; - new[[;;, {i+1, ip}]] = new[[;;, {ip, i+1}]]; - (* interchange contributes det(P)=-1. *) - val = -val; - ]; - (* Multiply with every other entry on the diagonal. *) - val = val * new[[i, i+1]]; - (*Build the Gauss vector*) - new[[i+2;;, i]] = new[[i+2;;, i]] / new[[i+1, i]]; - (* Update the remainder of the matrix using an outer product skew-symmetric update. Note that column and row i+1 are not affected by the update. *) - new[[i+2;;, i+2;;]] += (# - Transpose[#])& @ - Outer[Times, new[[i+2;;, i]], new[[i+2;;, i+1]]] - (* The above is much faster than this construct for me: Transpose[{new[[i+2;;,i]]}].{new[[i+2;;,i+1]]}-Transpose[{new[[i+2;;, i+1]]}].{new[[i+2;;,i]]}; *) - ]; - Return[val * new[[dim-1, dim]]] -] /; MatrixQ[mat, NumericQ] - -PfaffianLTL[mat_] := ( - Message[Pfaffian::number, mat]; - Return[Indeterminate] -) - (**** ****) @@ -167,8 +201,8 @@ PfaffianLTL[mat_] := ( HouseholderVector::usage = "HouseholderVector[vec] returns the Householder vector w of input vector vec; such that (I-2w w^\[Dagger]/w^\[Dagger]\[CenterDot]w) vec is a multiple of the unit vector Subscript[e, 1]." -HouseholderVector[vec_] := If[ - VectorQ[vec, NumberQ[#] && !MatchQ[#, _Complex] &], +HouseholderVector[vec_?VectorQ] := If[ + arrayRealQ[vec], HouseholderVectorReal[vec], HouseholderVectorComplex[vec] ] @@ -198,60 +232,10 @@ HouseholderVectorComplex[x_] := Module[ ] -SkewHouseholder::usage = "SkewHouseholder[mat] returns {T, Q} such that mat=Q T Q^T for a skew-symmetric matrix mat, where T is a skew-symmetric tridiagonal matrix and Q is a unitary matrix.\nFor real matrices, this should be the same as what is returned from the HessenbergDecomposition.\nFor the complex case, there is no Mathematica equivalent.\nThis full tridiagonalization is at the heart of the Pfaffian calculation in the Householder method." - -SkewHouseholder[mat_] := If[ - MatrixQ[mat, NumberQ[#] && !MatchQ[#, _Complex] &], - SkewHouseholderReal[mat], - SkewHouseholderComplex[mat] -] - -(* NOTE: These functions are not needed for the Pfaffian calculation, they are here merely for testing. *) -(* NOTE: These functions don't check (yet) if the input matrix is really skew-symmetric. *) - -SkewHouseholderReal[Mat_] := Module[ - {A, Q, v, beta, alpha}, - A = Mat; - Q = IdentityMatrix[Dimensions[A][[1]]]; - For[i = 1, i < Dimensions[A][[1]] - 1, i++, - (*Compute the Householder vector*) - {v, beta, alpha} = HouseholderVectorReal[A[[i + 1 ;; , i]]]; - (*eliminate the entries in row and column i*) - A[[i + 1, i]] = alpha; A[[i, i + 1]] = -alpha; A[[i + 2 ;;, i]] = 0; - A[[i, i + 2 ;;]] = 0; - (* update the matrix *) - w = beta* A[[i + 1 ;; , i + 1 ;;]] . v; - A[[i + 1 ;; , i + 1 ;; ]] += Transpose[{v}] . {w} - Transpose[{w}] . { v}; - (*accumulate the Householder reflections into the full transformation*) - y = Q[[1 ;;, i + 1 ;;]] . v; - Q[[1 ;; , i + 1 ;;]] -= beta*Transpose[{y}] . {v}]; - Return[{A, Q, One @ Length @ mat}] -] - -SkewHouseholderComplex[Mat_] := Module[ - {A, Q, v, beta, alpha}, - A = Mat; - Q = IdentityMatrix[Dimensions[A][[1]]]; - For[i = 1, i < Dimensions[A][[1]] - 1, i++, - (*Compute the Householder vector*) - {v, beta, alpha} = HouseholderVectorComplex[A[[i + 1 ;; , i]]]; - (*eliminate the entries in row and column i*) - A[[i + 1, i]] = alpha; A[[i, i + 1]] = -alpha; A[[i + 2 ;;, i]] = 0; - A[[i, i + 2 ;;]] = 0; - (*update the matrix*) - w = beta* A[[i + 1 ;; , i + 1 ;;]] . Conjugate[v]; - A[[i + 1 ;; , i + 1 ;; ]] += Transpose[{v}] . {w} - Transpose[{w}] . { v}; - (*accumulate the Householder reflections into the full transformation*) - y = Q[[1 ;;, i + 1 ;;]] . v; - Q[[1 ;; , i + 1 ;;]] -= beta*Transpose[{y}] . Conjugate[{v}]]; - Return[{A, Q, One @ Length @ mat}] -] - - PfaffianHouseholder::usage = "PfaffianHouseholder[mat] calculates the Pfaffian of skew-symmetric matrix mat by using the Householder tridiagonalization." PfaffianHouseholder[mat_] := If[ - MatrixQ[mat, NumberQ[#] && !MatchQ[#, _Complex] &], + arrayRealQ[mat], PfaffianHouseholderReal[mat], PfaffianHouseholderComplex[mat] ] @@ -263,11 +247,11 @@ PfaffianHouseholderReal[mat_] := Module[ If[OddQ[dim], Return[0]]; val = 1; For[i = 1, i < dim-1, i += 2, - (* Compute the Householder vector *) + (* Compute the Householder vector. *) {v, beta, alpha} = HouseholderVectorReal[new[[i+1 ;; , i]]]; (* multiply with off-diagonal entry and determinant of Householder reflection *) val *= If[beta == 0, 1, -1]*(-alpha); - (* update the matrix *) + (* Update the matrix *) w = beta * new[[i+1 ;; , i+1 ;;]] . v; new[[i+1 ;; , i+1 ;; ]] += Transpose[{v}] . {w} - Transpose[{w}] . {v} ]; @@ -281,17 +265,73 @@ PfaffianHouseholderComplex[mat_] := Module[ val = 1; If[OddQ[dim], Return[0]]; For[i = 1, i < dim-1, i += 2, - (*Compute the Householder vector*) + (* Compute the Householder vector. *) {v, beta, alpha} = HouseholderVectorComplex[new[[i+1 ;; , i]]]; (* multiply with off-diagonal entry and determinant of Householder reflection *) val *= If[beta == 0, 1, -1]*(-alpha); - (*update the matrix*) + (* Update the matrix. *) w = beta * new[[i+1 ;; , i+1 ;;]] . Conjugate[v]; new[[i+1 ;; , i+1 ;; ]] += Transpose[{v}] . {w} - Transpose[{w}] . {v} ]; Return[ val * new[[dim-1, dim]] ] ] + +SkewHouseholder::usage = "SkewHouseholder[mat] returns {T, Q} such that mat=Q T Q^T for a skew-symmetric matrix mat, where T is a skew-symmetric tridiagonal matrix and Q is a unitary matrix.\nFor real matrices, this should be the same as what is returned from the HessenbergDecomposition.\nFor the complex case, there is no Mathematica equivalent.\nThis full tridiagonalization is at the heart of the Pfaffian calculation in the Householder method." + +SkewHouseholder[mat_?MatrixQ] := If[ + arrayRealQ[mat], + SkewHouseholderReal[mat], + SkewHouseholderComplex[mat] +] + +(* NOTE: These functions are not needed for the Pfaffian calculation, they are here merely for testing. *) +(* NOTE: These functions don't check (yet) if the input matrix is really skew-symmetric. *) + +SkewHouseholderReal[mat_] := Module[ + {A, Q, v, beta, alpha}, + A = mat; + Q = IdentityMatrix[Dimensions[A][[1]]]; + For[i = 1, i < Dimensions[A][[1]] - 1, i++, + (*Compute the Householder vector*) + {v, beta, alpha} = HouseholderVectorReal[A[[i + 1 ;; , i]]]; + (*eliminate the entries in row and column i*) + A[[i + 1, i]] = alpha; + A[[i, i + 1]] = -alpha; + A[[i + 2 ;;, i]] = 0; + A[[i, i + 2 ;;]] = 0; + (* update the matrix *) + w = beta * A[[i + 1 ;; , i + 1 ;;]] . v; + A[[i + 1 ;; , i + 1 ;; ]] += Transpose[{v}] . {w} - Transpose[{w}] . {v}; + (*accumulate the Householder reflections into the full transformation*) + y = Q[[1 ;;, i + 1 ;;]] . v; + Q[[1 ;; , i + 1 ;;]] -= beta*Transpose[{y}] . {v} + ]; + Return[{A, Q, One @ Length @ mat}] +] + +SkewHouseholderComplex[mat_] := Module[ + {A, Q, v, beta, alpha}, + A = mat; + Q = IdentityMatrix[Dimensions[A][[1]]]; + For[i = 1, i < Dimensions[A][[1]] - 1, i++, + (*Compute the Householder vector*) + {v, beta, alpha} = HouseholderVectorComplex[A[[i + 1 ;; , i]]]; + (*eliminate the entries in row and column i*) + A[[i + 1, i]] = alpha; + A[[i, i + 1]] = -alpha; + A[[i + 2 ;;, i]] = 0; + A[[i, i + 2 ;;]] = 0; + (*update the matrix*) + w = beta* A[[i + 1 ;; , i + 1 ;;]] . Conjugate[v]; + A[[i + 1 ;; , i + 1 ;; ]] += Transpose[{v}] . {w} - Transpose[{w}] . {v}; + (*accumulate the Householder reflections into the full transformation*) + y = Q[[1 ;;, i + 1 ;;]] . v; + Q[[1 ;; , i + 1 ;;]] -= beta*Transpose[{y}] . Conjugate[{v}] + ]; + Return[{A, Q, One @ Length @ mat}] +] + (**** ****) @@ -303,31 +343,35 @@ PfaffianHouseholderComplex[mat_] := Module[ PfaffianHessenberg::usage = "PfaffianHessenberg[mat] calculates the Pfaffian of real skew-symmetric matrix mat by using the Hessenberg decomposition." -PfaffianHessenberg[mat_] := Module[ +(* NOTE: HessenbergDecomposition only receives a finite-preciesion matrix. *) +PfaffianHessenberg[mat_?arrayRealQ] := Module[ {H, Q}, {Q, H} = HessenbergDecomposition[N @ mat]; - Det[Q] * Apply[Times, Diagonal[H, 1][[1;;;;2]]] -] /; MatrixQ[mat, NumericQ[#] && !MatchQ[#, _Complex]&] -(* NOTE: HessenbergDecomposition only receives a finite-preciesion matrix. *) + Det[Q] * Apply[Times, Diagonal[H, 1][[1;; ;;2]]] +] PfaffianHessenberg[mat_] := ( Message[Pfaffian::Hessenberg]; - PfaffianHouseholderComplex[mat] -) (* otherwise *) + PfaffianLTL[mat] +) SkewHessenberg::usage = "SkewHessenberg[mat] tridiagonalize the real skew-symmetric matrix mat by using the Hessenberg decomposition.\nIt is an alias of HessenbergDecomposition." -SkewHessenberg[mat_] := Module[ - {H, Q}, - {Q, H} = HessenbergDecomposition[N @ mat]; - {H, Q, One[Length @ mat]} -] /; MatrixQ[mat, NumericQ[#] && !MatchQ[#, _Complex]&] - -SkewHessenberg[mat_] := ( - Message[SkewTridiagonalize::Hessenberg]; - SkewHouseholderComplex[mat] -) (* otherwise *) +SkewHessenberg[mat_] := Which[ + Not @ FreeQ[Normal @ mat, _Complex, {2}], (* complex numbers *) + Message[Skew::Hessenberg]; + SkewHouseholderComplex[mat], + FreeQ[Normal @ mat, _Real, {2}], (* exact numbers *) + Message[Skew::Hessenberg]; + SkewHouseholderReal[mat], + True, + Module[ + {H, Q}, + {Q, H} = HessenbergDecomposition[mat]; + {H, Q, One[Length @ mat]} + ] +] /; MatrixQ[mat, NumericQ] (**** ****) diff --git a/Q3/Kernel/Young.wl b/Q3/Kernel/Young.wl index a0e079446..877479045 100644 --- a/Q3/Kernel/Young.wl +++ b/Q3/Kernel/Young.wl @@ -70,13 +70,13 @@ PermutationForm[prm_?PermutationListQ] := DisplayForm @ RowBox @ { "(", Grid @ {Range[Length @ prm], prm}, ")" - } +} PermutationForm[expr_Association] := PermutationForm /@ expr PermutationForm[expr_] := ReplaceAll[ expr, { pp:$PermutationSpec :> PermutationForm[pp] } - ] +] Transposition::usage = "Transposition[{i, j}] returns Cycles[{{i, j}}], representing the transposition of elements i and j. It is a shortcut to Cycles[{{i, j}}].\nTransposition[n] is equivalent to Transposition[{n, n+1}]." @@ -114,6 +114,8 @@ FerrersDiagram[yt:{{___Integer}..}] := FerrersDiagram[YoungShape @ yt] FerrersDiagram[yt_YoungTableau] := FerrersDiagram[YoungShape @ yt] +(**** ****) + YoungShapeQ::usage="YoungShapeQ[shape] returns True if and only if shape is an integer partition, arranged in non-increasing order.\nA Young shape defines a Young diagram." YoungShapeQ[YoungShape[data_List]] := YoungShapeQ[data] @@ -124,6 +126,23 @@ YoungShapeQ[pp:{__Integer?NonNegative}] := Apply[GreaterEqual, pp] (* NOTE: Must allow 0 since some functions uses 0 in shape specification. *) +YoungShape::usage = "YoungShape[tb] returns the shape, i.e., the integer partition of Young tableau tb." + +Format[ys:YoungShape[data:{___}]] := + Interpretation[YoungDiagram @ data, ys] + +YoungShape[YoungTableau[tb_]] := YoungShape[tb] + +YoungShape[tb:{__List}] := YoungShape[Length /@ tb] + + +YoungShapes::usage = "YoungShapes[n] returns the list of all possible Young shapes for integern n." + +YoungShapes[n_Integer] := YoungShape /@ IntegerPartitions[n] + +(**** ****) + + (**** ****) YoungTrim::usage="YoungTrim[shape] trims trailing zeros from shape.\nYoungTrim[tbl] trims trailing {}s from Young tableau tbl." @@ -153,30 +172,11 @@ YoungTranspose[YoungTableau[data_]] := YoungTableau[YoungTranspose @ data] YoungTranspose[tb:{{___Integer}..}] := With[ { new = YoungTranspose[Length /@ tb] }, Table[tb[[j, i]], {i, Length @ new}, {j, new[[i]]}] - ] +] (* NOTE: tb does not need to be a semi-standard Young tableau. Any Young-like tableau is allowed. This is useful in Schur transform. *) -(**** ****) - -YoungShape::usage = "YoungShape[tb] returns the shape, i.e., the integer partition of Young tableau tb." - -Format[ys:YoungShape[data:{___}]] := - Interpretation[YoungDiagram @ data, ys] - -YoungShape[YoungTableau[tb_]] := YoungShape[tb] - -YoungShape[tb:{__List}] := YoungShape[Length /@ tb] - - -YoungShapes::usage = "YoungShapes[n] returns the list of all possible Young shapes for integern n." - -YoungShapes[n_Integer] := YoungShape /@ IntegerPartitions[n] - -(**** ****) - - YoungTableauCount::usage = "YoungTableauCount[shape] uses the hook length formula to count the number of standard Young tableaux of 'shape'.\nYoungTableauCount[n] gives the total number of standard Young tableaux for all partitions of integer 'n'.\nBorrowed from NumberOfTableaux in Combinatorica package." YoungTableauCount[YoungShape[pp_]] := YoungTableauCount[pp] @@ -189,8 +189,8 @@ YoungTableauCount[pp_?YoungShapeQ] := Module[ pp[[j]] + qq[[k]] - j - k + 1, {j, Length[pp]}, {k, pp[[j]]} - ] - ] + ] +] YoungTableauCount[n_Integer] := Total @ Map[YoungTableauCount, IntegerPartitions @ n] @@ -209,12 +209,12 @@ GroupRegularRepresentation[grp_, elm_] := Transpose @ GroupMultiplicationTable[grp], GroupElementPosition[grp, elm] ] - ] /; GroupElementQ[grp, elm] + ] /; GroupElementQ[grp, elm] GroupRegularRepresentation[grp_, elm_] := ( Message[GroupRegularRepresentation::elm, grp, elm]; One[GroupOrder @ grp] - ) +) (* For the regular representation of the symmetric group *) theKetFormatQ[_Cycles] = True @@ -290,11 +290,11 @@ characterSymmetricGroup[shape_?YoungShapeQ, class_?YoungShapeQ] := {j, 1, Length[shape]}, {i, 1, shape[[j]]} ] - ] /; And[ - Total[shape] == Total[class], - Length[class] >= 1, - First[class] > 1 - ] + ] /; And[ + Total[shape] == Total[class], + Length[class] >= 1, + First[class] > 1 + ] (**** ****) @@ -315,12 +315,12 @@ SymmetricGroupClasses::uasge = "SymmetricGroupClasses[n] returns an association SymmetricGroupClasses[n_Integer] := With[ { elm = GroupElements[SymmetricGroup @ n] }, Reverse @ KeySort[GroupBy[elm, theIrreducibleLabel[n]], LexicographicOrder] - ] +] theIrreducibleLabel[n_Integer][Cycles[spec_List]] := With[ { body = ReverseSort[Length /@ spec] }, Join[body, Table[1, n-Total[body]]] - ] +] (**** ****) @@ -933,7 +933,7 @@ KetSymmetrize[expr_, tbl_?YoungTableauQ] := Module[ } /. { w_Ket :> Total[Permute[w[[tt]], #]& /@ aa] } - ] +] (**** ****) @@ -948,7 +948,7 @@ InversionVector[p_?PermutationListQ] := Module[ { new = Take[p, #]& /@ InversePermutation[p] }, Most @ MapThread[ Function[{x, y}, Count[x, _?(# > y &)]], {new, Range[Length @ p]}] - ] /; (Length[p] > 0) +] /; (Length[p] > 0) (**** ****) @@ -963,6 +963,7 @@ AdjacentTranspositions[{}] = {} AdjacentTranspositions[cyc_Cycles] := AdjacentTranspositions[PermutationList @ cyc] + intoTranspositions::usage="intoTranspositions[perm] represents perm as product of transpositions of immediate neighbors. An entry value of k in the returned list denotes the transposition (k,k+1)." intoTranspositions[cyc_Cycles] := @@ -981,7 +982,7 @@ intoTranspositions[prm_?PermutationListQ] := Module[ ] ]; Return[trs] - ] +] fromTranspositions::usage = "fromTranspositions[trs] is the inverse operation of intoTranspositions.\nGiven here just for heuristic reasons." @@ -1002,7 +1003,7 @@ YoungDistance[{x_Integer, y_Integer}, data_?YoungTableauQ] := Module[ { xx = FirstPosition[data, x], yy = FirstPosition[data, y] }, Dot[yy - xx, {-1, 1}] - ] +] (**** ****) @@ -1020,7 +1021,7 @@ BratteliDiagram[n_Integer, opts : OptionsPattern[Graph]] := Module[ GraphLayout -> "LayeredEmbedding", VertexLabels -> vtx, VertexLabelStyle -> Large ] - ] +] brattelli[shape_?YoungShapeQ] := Map[UndirectedEdge[shape, #] &, PileYoungShape[shape]] @@ -1035,9 +1036,11 @@ PileYoungShape[shape_?YoungShapeQ] := Module[ { new = Append[YoungTrim[shape], 0], k }, new = Table[ - ReplacePart[new, k -> new[[k]]+1], {k, 1, Length[new]}]; + ReplacePart[new, k -> new[[k]]+1], + {k, 1, Length[new]} + ]; YoungTrim /@ Select[new, YoungShapeQ] - ] +] (**** ****) @@ -1051,7 +1054,7 @@ YoungInvariantBasis[bs:{Ket[__?YoungTableauQ] ..}] := Module[ op }, op = Total @ GroupElements @ SymmetricGroup[n]; DeleteCases[Union @ KetCanonical[op ** bs], 0] - ] +] (**** ****) diff --git a/Q3/PacletInfo.wl b/Q3/PacletInfo.wl index 348e28343..4c5e100d4 100644 --- a/Q3/PacletInfo.wl +++ b/Q3/PacletInfo.wl @@ -1,6 +1,6 @@ Paclet[ "Name" -> "Q3", - "Version" -> "3.4.3", + "Version" -> "3.4.6", "WolframVersion" -> "12.3+", "Updating" -> Automatic, "Loading" -> "Startup", diff --git a/Q3/PlaybookTools/Kernel/PlaybookTools.wl b/Q3/PlaybookTools/Kernel/PlaybookTools.wl index d4aa586f7..fdfc19111 100644 --- a/Q3/PlaybookTools/Kernel/PlaybookTools.wl +++ b/Q3/PlaybookTools/Kernel/PlaybookTools.wl @@ -118,8 +118,8 @@ fileDeploy[src_String, dst_String, OptionsPattern[PlaybookDeploy]] := Module[ Return[$Failed] ]; - Print["\tSetting the stylesheet to \"PlaybookX.nb\" for printing (if at all) ..."]; - SetOptions[nb, StyleDefinitions -> "PlaybookX.nb"]; (* for printing if at all *) + Print["\tSetting the stylesheet to \"PlaybookNub.nb\" for printing (if at all) ..."]; + SetOptions[nb, StyleDefinitions -> "PlaybookNub.nb"]; (* for printing if at all *) Print["\tSetting the banner as \"", $PlaybookBanner, "\" for printing (if at all) ..."]; SetBanner[nb, $PlaybookBanner]; (* for printing if at all *) diff --git a/RELEASES.md b/RELEASES.md index 9eb9f266f..adbe99bc0 100644 --- a/RELEASES.md +++ b/RELEASES.md @@ -1,5 +1,22 @@ # Release Notes +## 3.4.6 + +- Further performance improvement for fermionic quantum computing +- Style sheet PlaybookX has been renamed PlaybookNub. + +## 3.4.5 + +- Pfaffian has greatly improved; almost 10 times faster. This also improves the performance of WickState, WickUnitary, WickExpectation, WickGreensFunction, WickLogarithmicNegativity, WickEntanglementEntropy, etc. +- New functions: ChebyshevApproximation, ChebyshevCoefficients +- Matrix and Elaborate support WickState, WickUnitary, WickOperator, Measurement[_?FermionQ]. + +## 3.4.4 + +- New functions: NambuMatrix, NambuRandomMatrix, PauliMatrixQ, CliffordMatrixQ. +- Improved: Pfaffian, SkewTridiagonalize, DickeBasis, PauliQ. +- QuantumCircuit supports WickUnitary, WickOperator, Measurement[fermions] and some other fermionic quantum computing elements. + ## 3.4.3 - New function WickEntanglementEntropy (cf. EntanglementEntroy) is dedicated to the context of Wick state.