Multiparticle quantum walks for distinguishing hard graphs

TL;DR

This paper explores how multi-particle quantum walks can be used to distinguish complex graphs, including those that classical methods struggle with, by providing theoretical proofs and empirical evidence.

Contribution

It introduces quantum walk-based methods for graph isomorphism testing that outperform classical WL tests on certain hard graph classes.

Findings

k-QW with superposition states distinguishes k-CFI graphs

k-1 QW with localized states distinguishes k-CFI graphs

Theoretical proofs support quantum walk effectiveness on complex graphs

Abstract

Quantum random walks have been shown to be powerful quantum algorithms for certain tasks on graphs like database searching, quantum simulations etc. In this work we focus on its applications for the graph isomorphism problem. In particular we look at how we can compare multi-particle quantum walks and well known classical WL tests and how quantum walks can be used to distinguish hard graphs like CFI graphs which k-WL tests cannot distinguish. We provide theoretical proofs and empirical results to show that a k-QW with input superposition states distinguishes k-CFI graphs. In addition we also prove that a k-1 QW with localized input states distinguishes k-CFI graphs. We also prove some additional results about strongly regular graphs (SRGs).