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

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 N2-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.Program summaryProgram Title: QSWalk.mProgram Files doi:http://dx.doi.org/10.17632/8rwd3j9zhk.1Licensing provisions: GNU General Public License 3 (GPL)Programming language: Mathematica.Nature of problem: The QSWalk package provides a method for simulating quantum stochastic walks on arbitrary (directed/undirected, weighted/unweighted) graphs.Solution method:For an N-vertex graph, the solution of a quantum stochastic walk can be expressed as an N2×N2 sparse matrix exponential. The QSWalk package makes use of Mathematica’s sparse linear algebra routines to solve this efficiently.Restrictions:The size of graphs that can be treated is constrained by available memory.