SQUWALS: A Szegedy QUantum WALks Simulator

TL;DR

This paper introduces SQUWALS, a memory-efficient classical simulator for Szegedy's quantum walk, enabling error-free testing of quantum algorithms like quantum PageRank with scalable performance.

Contribution

It presents a novel memory-saving algorithm for simulating Szegedy's quantum walk and extends it to mixed states and semiclassical variants, implemented in the SQUWALS Python package.

Findings

Scales as O(N^2) in time and memory for graph size N

Supports simulation of mixed states and semiclassical walks

Enables high-level applications like quantum PageRank

Abstract

Szegedy's quantum walk is an algorithm for quantizing a general Markov chain. It has plenty of applications such as many variants of optimizations. In order to check its properties in an error-free environment, it is important to have a classical simulator. However, the current simulation algorithms require a great deal of memory due to the particular formulation of this quantum walk. In this paper we propose a memory-saving algorithm that scales as $\mathcal{O}(N^2)$ with the size $N$ of the graph. We provide additional procedures for simulating Szegedy's quantum walk over mixed states and also the Semiclassical Szegedy walk. With these techniques we have built a classical simulator in Python called SQUWALS. We show that our simulator scales as $\mathcal{O}(N^2)$ in both time and memory resources. This package provides some high-level applications for algorithms based on Szegedy's quantum walk, as for example the quantum PageRank.