This Mathematica package defines some basic function to ease the handling and simulation of discrete-time one-dimensional quantum walks with two-dimensional coins.
The QM package is required for proper functioning.
The easiest and recommended way is to clone (via git clone) this repository into the Applications directory (the one given by Mathematica upon evaluation of FileNameJoin @ {$UserBaseDirectory, "Applications"}).
Once the package is installed, it needs to be imported with Needs["QuantumWalks`"].
Alternatively, the package can be imported directly from GitHub without installing it, using
Get["https://raw.githubusercontent.com/lucainnocenti/QuantumWalks/master/QuantumWalks.m"];
QWStepEvolutionMatrix[3, HadamardMatrix[2]] // MatrixForm
The function QWCoinMatrix can also be used as a transparent wrapper to clarify when an input should be understood as the matrix representing a coin operation.
E.g. in the example above we can replace HadamardMatrix[2] with QWCoinMatrix[HadamardMatrix[2]].
We can get the evolution matrix corresponding to multiple QW steps using QWManyStepsEvolutionMatrix. For example, to compute the matrix describing two steps each one using a Hadamard coin operation, with a total number of sites in the underlying space of 3, we can use:
QWManyStepsEvolutionMatrix[3,
Table[QWCoinMatrix[HadamardMatrix@2], 2]
] // MatrixForm
To evolve an input state on a fixed single walker position, and initial coin state either H or V, we can use
numberOfSteps = 4;
evolutionMatrix = QWManyStepsEvolutionMatrix[
numberOfSteps + 1,
Table[QWCoinMatrix[RandomUnitary@2], numberOfSteps]
];
outputState1Up = evolutionMatrix[[All, 1]] // Chop (* output entering with |1,H> *)
outputState1Down = evolutionMatrix[[All, 2]] // Chop (* output entering with with |1,V> *)Note that this requires the function RandomUnitary to generate random unitary evolution. This is defined e.g. in QM.
See QuantumWalksDemonstrations.nb for other usage examples.