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 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. 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 N 2 ×N 2 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. Running time:Running time depends on the size of the graph and the propagation time of the walk.

Zero transfer in continuous-time quantum walks

“…To simulate environmental interactions decoherence was introduced to the system in the form of scattering, dissipation and dephasing through the use of the Lindblad operators, L k , in QSWs. This was done using the 'QSWalk' package by Falloon et al [29], which computes quantum stochastic walks on graphs using the Mathematica system. Figure 13 shows the probability distribution once again for C 4 with the same complex edge weights given in the example before with decoherence introduced to the system.…”

QSWalk.jl: Julia package for quantum stochastic walks analysis

“…The model is used to study quantum-to-classical transition and to probe the influence of quantum evolution on the efficiency of information processing [3]. Quantum stochastic walks have been widely analysed in context of propagation [3,4,5,6,7], application in computer science [8,9,10] and physics [11,12,13]. In particular, new model of fast-propagating quantum walk on arbitrary directed graph was developed [4,6].…”

“…Comparison with the existing software. Many software packages for simulating quantum computing [15], including some focused on various models of discrete quantum walks [16,17] and continuous-time quantum walks [9,18] are available. Most of them are focused on the discrete-time evolution, restricted to the unitary operations and undirected graphs.…”

“…Most of them are focused on the discrete-time evolution, restricted to the unitary operations and undirected graphs. The first exception is QSWalk.m package [9] for Wolfram Mathematica, which provides implementation of a quantum stochastic walk model. Package QSWalk.m utilizes MatrixExp function and sparse representation of superoperators for simulating the evolution of quantum stochastic walks.…”

Experimental Parity-Time Symmetric Quantum Walks for Centrality Ranking on Directed Graphs

“…One promising application of quantum walks is to provide an efficient quantum algorithm for vertex centrality ranking in network analysis [6,[33][34][35][36][37]. Using continuous-time quantum walks (CTQWs), the aforementioned algorithm has been implemented on an undirected graph [10].…”