From 54a0b59137ddc9dd73e0a0c900edb6793a359965 Mon Sep 17 00:00:00 2001 From: Yuning Date: Tue, 2 Jul 2024 20:32:20 +0200 Subject: [PATCH] Update docs and Project.toml. New version 0.2.0 :fire:. --- Project.toml | 2 +- docs/src/assets/example1.png | Bin 0 -> 87683 bytes docs/src/assets/example2.png | Bin 0 -> 118220 bytes docs/src/guide.md | 198 +++++++++++++++++++++++++++++++---- docs/src/index.md | 33 +++++- docs/src/manual.md | 3 + 6 files changed, 212 insertions(+), 24 deletions(-) create mode 100644 docs/src/assets/example1.png create mode 100644 docs/src/assets/example2.png diff --git a/Project.toml b/Project.toml index 3221b42..5bcb590 100644 --- a/Project.toml +++ b/Project.toml @@ -1,7 +1,7 @@ name = "SpinShuttling" uuid = "eff49e45-ae6d-44ae-9170-2124a87971c9" authors = ["Yuning Zhang"] -version = "0.1.7" +version = "0.2.0" [deps] LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" diff --git a/docs/src/assets/example1.png b/docs/src/assets/example1.png new file mode 100644 index 0000000000000000000000000000000000000000..bc97abf9bed399f83e44bc96f617a86c4b8ccc2f GIT binary patch literal 87683 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z?0X~)9*TkX#V$C6pyymAP0qUY)rFl`9xVl!3Ymbb`JuI|S9^UunD7cE6zzWABqhJ1 zZ}s$@0>rAL?}jquAO(Kg%L9VA8tnKpbuS6Zt61Lxyi#&v!GBbfOMKU^W`cQ>P0VYv zWmG8s(R654iiw_{-f_Uzd%woPagb%_h2ua30iq6pMx6uo)@F`AN^DgW2t`Qy&97WH z03Zz3CFas4(AaI}0fB*lZfe}xMEueyN7Hh0c6{C@W?gBj#kl1#$|a)55w$=x{j-II z1AjREsvjt^yX+ ztc+uIadSfmY=`9{=$Fm%PuF5W;3R>BbW6ij33LK{MeWK;cmcbT05k*5rOx0KmeM?TuSO99hM(AGut@PWe;dmnK6 z^l1sJJL3R)J&!(nix|7b=L=&CehOeqB~WhYDEFxU3?>aU3no>-_uR<#Lr3utI$tm*B3RW_S3`NwH8>NFod7-xwEyVf6`H%7eX45i+?htyudS_>p0_*D zwloj}P}5o#N!6|->hHSUb7IVT1W+?naP%euLRMCTTPa>#nY`w`V`Db^_U!|~4{Ya6 zHA^US+pAY`adB9Z_U7hI!ooV7%MZn({TK!==4NKZH|0@i8bLCG2aQEX@kjY^8IbNZJODQrb5m1tO3F%kC#gz)h9%90sCnMt``s|wgC1Qqe z81aS_a#6s^ZrrFV4&{W@=FKk`z9aIDVsKCm=vfn!m*vGiWMp2;`5}netl;;hpD@Vr z7l+={Znek?LcAjQ2yP)k9RTVyG&OOfxqAV-%%LA^!a)skRN0e}1;hiyCEGuRVCFp~0`dNY@98CN%+5$SNd*}Uf_Tf#&o^Tl z!047Q@30>ikrOKBaj~$lu(Qu`jIO4_H*&ByMY03(14yZe@v zr6t>j4Pqy8t8Nw$xas|IB}5l$@#wcBBa__t+fw6CRiR7q@y_L4qds#0Nc$h*^mJMu zzIP8+sQb6$ffDbSc(DKNKZkFLF`vEcF1vSSJ~iGO6E&uI7JvErHP#cA*Y7_!0-XFZ zd^!{k>tj{YiRYNq#&dnYa^j8pvRNefn&-$Jnqp<{gM1 zl)4AINo&xG7qo_WI>QeEfDwV_y#f+iT(jN*-RFj~T(KZT{8b!+K8Bs{W5nQ`3Te-?Mk{QSIvKVIgrwet3aYuA9yzA5{$1+$m9=6hGVTVZkU$`L<{jO>0o@qPd8HT!@0=YKu9 YAwuul3-u$%h{vR>X`qp>ZWsLj0k+LE +Model for spin shuttling +Spin Number: n=1 +Initial State: |Ψ₀⟩=[0.707, 0.707] +Noise Channel: OrnsteinUhlenbeckField(0, [0.05, 10.0], 0.07071067811865475) +Time Discretization: N=301 +Process Time: T=400 +Shuttling Paths: + ┌──────────────────────────────┐ + 10 │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋│ x1(t) + │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀│ + │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀│ + │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ + │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ + │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ + │⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ + │⠀⠀⠀⣀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ + 0 │⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ + └──────────────────────────────┘ + ⠀0⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀400⠀ + The fidelity of the spin state after shuttling can be calculated using numerical integration of the covariance matrix. + +This provides us an overview of the model. It's a single spin shuttling problem with initial state `Ψ₀` and an Ornstein-Uhlenbeck noise. The total time of simulation is `T`, which is discretized into `N` steps. + +The state fidelity after such a quantum process can be obtained by different numerical methods. ```julia -f1=averagefidelity(model) -``` -The fidelity can also be obtained from Monte-Carlo sampling. -```julia -f2, f2_err=sampling(model, fidelity, M) +f1=averagefidelity(model); # direct integration + +f2, f2_err=sampling(model, fidelity, M); # Monte-Carlo sampling ``` For the single spin shuttling at constant velocity, analytical solution is also available. ```julia -f3=1/2*(1+W(T,L,B)) +f3=1/2*(1+W(T,L,B)); ``` -We can compare the results form the three methods. +We can compare the results form the three methods and check their consistency. ```julia @assert isapprox(f1, f3,rtol=1e-2) @assert isapprox(f2, f3, rtol=1e-2) +println("NI:", f1) +println("MC:", f2) +println("TH:", f3) +``` +``` +NI:0.5172897445804854 +MC:0.5210640948921192 +TH:0.5183394145238882 +``` +The pure dephasing channel is computationaly simple, and can be represented by a dephasing matrix $w$, such that the final density state after the channel is given by $\mathcal{E}(\rho)=w \odot\rho$. Here $\odot$ is a element-wise Hadmard product. +```julia +Ψ= model.Ψ +ρ=Ψ*Ψ' +w=dephasingmatrix(model) +w.*ρ +``` +The final density matrix is shown in output. +``` +2×2 Matrix{Float64}: + 0.5 0.0172897 + 0.0172897 0.5 +``` +We can check that the fidelity between the initial and final state is consistent with results above. +```julia +f=(Ψ'*ρt*Ψ) +``` + +``` +0.5172897445804852 +``` + +## Dephasing of entangled spin pairs during shuttling. +Following the approach above, we can further explore the multi-spin system. +The general abstraction on such a problem is given by the data type `ShuttlingModel`. +```julia +ShuttlingModel(n, Ψ, T, N, B, X, R) +``` +User can freely define a n-qubit system with arbitrary initial state. Here, `X=[x1,x2...]` is an array of function, containing spin trajectories $x_i(t)$. `R` is a random function constructed from the specific noise process. + +One more example is the shuttling of two spin pairs. We can define such a two spin system. +```julia +L=10; σ =sqrt(2)/20; M=5000; N=501; T1=100; T0=25; κₜ=1/20; κₓ=1/0.1; +B=OrnsteinUhlenbeckField(0,[κₜ,κₓ],σ) +model=TwoSpinModel(T0, T1, L, N, B) +``` +

+Model for spin shuttling
+Spin Number: n=2
+Initial State: |Ψ₀⟩=[0.0, 0.707, -0.707, 0.0]
+Noise Channel: OrnsteinUhlenbeckField(0, [0.05, 10.0], 0.07071067811865475)
+Time Discretization: N=501
+Process Time: T=125
+Shuttling Paths:
+      ┌──────────────────────────────┐      
+   10 │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⡹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ x1(t)
+      │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ x2(t)
+      │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│      
+      │⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│      
+      │⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│      
+      │⠀⠀⠀⠀⠀⡴⠋⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│      
+      │⠀⠀⠀⣠⠞⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│      
+      │⠀⢀⡜⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│      
+    0 │⣴⣋⣀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│      
+      └──────────────────────────────┘      
+      ⠀0⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀200⠀      
+
+The system is initialized in the Bell state $\ket{\Psi^-}$. +The model encapsulated a model of two spin shuttled in a sequential manner, as we can see from the two trajectories `x1(t)` and `x2(t)`. One spin goes first and then follows another, with waiting time `T0`. This is modeled by the piece-wise linear trajectories. +We can see some quite interesting covariance from such a system. +```julia +heatmap(collect(model.R.Σ), title="cross covariance matrix, two spin EPR") +``` +![Covariance Matrix](assets/example2.png) + +We can check that the dephasing of the system and calculate its fidelity as before. +```julia +f1=averagefidelity(model) +f2, f2_err=sampling(model, fidelity, M) +f3=1/2*(1+W(T0, T1, L,B)) + println("NI:", f1) println("MC:", f2) println("TH:", f3) ``` -## Shuttling of entangled spin pairs. \ No newline at end of file +``` +NI:0.970066663028708 +MC:0.9699908129499957 +TH:0.9700818676163352 +``` +The density matrix after the channel can be given by the dephasing matrix. +```julia +Ψ= model.Ψ +ρ=Ψ*Ψ' +w=dephasingmatrix(model) + +ρt=w.*ρ +``` + +``` +4×4 Matrix{Float64}: + 0.0 0.0 -0.0 0.0 + 0.0 0.5 -0.470067 0.0 + -0.0 -0.470067 0.5 -0.0 + 0.0 0.0 -0.0 0.0 +``` \ No newline at end of file diff --git a/docs/src/index.md b/docs/src/index.md index 823f272..00de54a 100644 --- a/docs/src/index.md +++ b/docs/src/index.md @@ -3,10 +3,35 @@ *Simulate the multiple-spin shuttling problem under correlated stochastic noise.* ## Installation - -SpinShuttling can be installed using the Julia package manager. + +`SpinShuttling.jl` can be installed by cloning the repository from github. +```shell +git clone https://github.com/EigenSolver/SpinShuttling.jl.git +``` +Go to the directory of the project in terminal. +```shell +cd ./SpinShuttling.jl +``` From the Julia REPL, type `]` to enter the Pkg REPL mode and run - +```julia +pkg> add . ``` + + +## What does this package do +This package provides a set of abstractions and numericals tools to simulating dynamics of multi-spin system under correlated noises, based on the Gaussian random field approach. + + +While we provided specially optimized models for spin shuttling problems. This package can also be used to simulate more general *correlated open-quantum dynamics*. + +The following two approaches are supported. +- Direct numerical integration for pure dephasing. +- Monte-Carlo sampling for open-system dynamics. + +## About spin shuttling +Spin shuttling has recently emerged as a pivotal technology for large-scale semiconductor quantum computing. By transporting qubits between quantum dots, spin shuttling enables entanglement between non-neighboring qubits, which is essential for quantum error correction. However, the spin qubit becomes decohered by magnetic noise during the shuttling process. Since the noise varies in time and space in a correlated manner, the associated dephasing in a system of several entangled spins often cannot be treated using the standard theory of random processes and requires more advanced mathematical instruments. +In our latest work, we employ the Gaussian random field (GRF) to model the magnetic noise varying in both space and time. By projecting trajectories of spin qubits onto the random field, the correlated noises experienced by multi-spin system can be effectively captured, enabling further study on spin dynamics, dephasing and quantum information applications. diff --git a/docs/src/manual.md b/docs/src/manual.md index daabd5d..f1bbc75 100644 --- a/docs/src/manual.md +++ b/docs/src/manual.md @@ -23,6 +23,9 @@ TwoSpinModel TwoSpinParallelModel ``` +```@docs +ShuttlingModel +``` ```@docs dephasingmatrix