Graph isomorphism and Gaussian boson sampling

TL;DR

This paper establishes a novel link between Gaussian boson sampling, a quantum computing approach, and the graph isomorphism problem, proposing quantum measurement-based invariants for graph comparison.

Contribution

It introduces a quantum-based method to determine graph isomorphism using photon detection probabilities in Gaussian boson sampling.

Findings

Photon detection probabilities form complete graph invariants.

The method distinguishes non-isomorphic graphs with identical spectra.

Numerical simulations validate the approach on complex graph families.

Abstract

We introduce a connection between a near-term quantum computing device, specifically a Gaussian boson sampler, and the graph isomorphism problem. We propose a scheme where graphs are encoded into quantum states of light, whose properties are then probed with photon-number-resolving detectors. We prove that the probabilities of different photon-detection events in this setup can be combined to give a complete set of graph invariants. Two graphs are isomorphic if and only if their detection probabilities are equivalent. We present additional ways that the measurement probabilities can be combined or coarse-grained to make experimental tests more amenable. We benchmark these methods with numerical simulations on the Titan supercomputer for several graph families: pairs of isospectral nonisomorphic graphs, isospectral regular graphs, and strongly regular graphs.