A Quantum-Inspired Algorithm for Graph Isomorphism

TL;DR

This paper evaluates a quantum-inspired classical algorithm for graph isomorphism, inspired by a photonic quantum approach, providing a new necessary condition and analyzing its effectiveness compared to existing algorithms.

Contribution

It introduces a classical algorithm based on quantum sampling properties to test graph isomorphism, inspired by a quantum algorithm, and analyzes its relation to existing methods.

Findings

The classical algorithm can identify non-isomorphic graphs using quantum sampling properties.

The analysis compares the quantum-inspired method with classical algorithms like color refinement and Babai's algorithm.

The work highlights limitations and potential of quantum-inspired classical approaches for graph isomorphism.

Abstract

The Noisy Intermediate-Scale Quantum (NISQ) era of technology in which we currently find ourselves is defined by non-universality, susceptibility to errors and noise, and a search for useful applications. While demonstrations of practical quantum advantage remain elusive in this era, it provides space to develop and analyze the advantages and limitations of systems and their ability to solve problems. In this work, we critically assess a proposed quantum algorithm for the graph isomorphism problem, implemented on a photonic quantum device. Inspired by the nature of this quantum algorithm, we formulate a necessary condition for the isomorphism of graphs encoded in Gaussian boson samplers and a classical algorithm to test for it. Our classical algorithm makes use of efficiently computable statistical properties of a quantum sampling system to show a pair of graphs fail to meet our necessary condition and thus cannot be isomorphic. We analyze our algorithm in the context of the inspiring, sampler-based quantum algorithm of Br\`adler et. al., the classical color refinement algorithm, and the state-of-the-art quasi-polynomial Babai algorithm.