A Quantum Observable for the Graph Isomorphism Problem

TL;DR

This paper introduces a quantum observable designed to determine graph isomorphism with certainty for isomorphic graphs and high probability for non-isomorphic graphs, based on a specific Hilbert space construction.

Contribution

It proposes a novel quantum observable in a complex Hilbert space that distinguishes isomorphic from non-isomorphic graphs, advancing quantum approaches to graph isomorphism.

Findings

Returns 'yes' with certainty for isomorphic graphs

Returns 'no' with high probability for non-isomorphic graphs

The implementability efficiency remains uncertain

Abstract

Suppose we are given two graphs on $n$ vertices. We define an observable in the Hilbert space $\Co[(S_n \wr S_2)^m]$ which returns the answer ``yes'' with certainty if the graphs are isomorphic and ``no'' with probability at least $1-n!/2^m$ if the graphs are not isomorphic. We do not know if this observable is efficiently implementable.