QSWalk: a Mathematica package for quantum stochastic walks on arbitrary graphs

TL;DR

QSWalk is a Mathematica package that efficiently simulates quantum stochastic walks on arbitrary graphs, blending quantum and classical dynamics using Lindblad formalism.

Contribution

It introduces a versatile Mathematica tool for simulating quantum stochastic walks on arbitrary graphs, combining quantum and classical behaviors within a unified framework.

Findings

Demonstrates efficient simulation of QSWs on various graphs.

Shows the package's capability to handle directed and weighted graphs.

Provides example case studies illustrating the package's functionality.

Abstract

We present a Mathematica package, QSWalk, to simulate the time evaluation of Quantum Stochastic Walks (QSWs) on arbitrary directed and weighted graphs. QSWs are a generalization of continuous time quantum walks that incorporate both coherent and incoherent dynamics and as such, include both quantum walks and classical random walks as special cases. The incoherent component allows for quantum walks along directed graph edges. The dynamics of QSWs are expressed using the Lindblad formalism, originally developed for open quantum systems, which frames the problem in the language of density matrices. For a QSW on a graph of $N$ vertices, we have a sparse superoperator in an $N^2$-dimensional space, which can be solved efficiently using the built-in MatrixExp function in Mathematica. We illustrate the use of the QSWalk package through several example case studies.