Combinatorics

Discrete mathematics

Physics

Graph

Pure mathematics

Voltage graph

Mathematics

Graph isomorphism

Crystallography

Crystal structure

Gaussian

Quantum graph

Chemistry

Isomorphism (crystallography)

Line graph

Isospectral

Quantum mechanics

Graph homomorphism

Special Matrices

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.

Graph isomorphism and Gaussian boson sampling