Decoherence enhances performance of quantum walks applied to graph isomorphism testing

TL;DR

This paper introduces a quantum algorithm for graph isomorphism testing that performs optimally in a partially coherent regime, leveraging decoherence to enhance the ability to distinguish non-isomorphic graphs through quantum stochastic walks.

Contribution

The study demonstrates that decoherence can improve quantum walk-based graph isomorphism testing, providing a novel approach that exploits partial coherence for better performance.

Findings

Decoherence enhances the distinguishing power of quantum walks.

QSW-based graph invariants can identify non-isomorphic graphs only distinguishable with decoherence.

The algorithm outperforms purely coherent or classical methods in specific cases.

Abstract

Computational advantages gained by quantum algorithms rely largely on the coherence of quantum devices and are generally compromised by decoherence. As an exception, we present a quantum algorithm for graph isomorphism testing whose performance is optimal when operating in the partially coherent regime, as opposed to the extremes of fully coherent or classical regimes. The algorithm builds on continuous-time quantum stochastic walks (QSWs) on graphs and the algorithmic performance is quantified by the distinguishing power (DIP) between non-isomorphic graphs. The QSW explores the entire graph and acquires information about the underlying structure, which is extracted by monitoring stochastic jumps across an auxiliary edge. The resulting counting statistics of stochastic jumps is used to identify the spectrum of the dynamical generator of the QSW, serving as a novel graph invariant, based on which non-isomorphic graphs are distinguished. We provide specific examples of non-isomorphic graphs that are only distinguishable by QSWs in the presence of decoherence.