Quantum and non-signalling graph isomorphisms

TL;DR

This paper introduces quantum and non-signalling graph isomorphisms via a nonlocal game framework, establishing their properties, relationships, and computational complexities, including the operational interpretation of fractional isomorphism and the existence of non-isomorphic yet quantum isomorphic graphs.

Contribution

It defines quantum and non-signalling graph isomorphisms, relates non-signalling isomorphism to fractional isomorphism, and demonstrates the distinction and undecidability of quantum isomorphism frameworks.

Findings

Non-signalling isomorphism equals fractional isomorphism.

Quantum isomorphism corresponds to solutions of polynomial systems in non-commuting variables.

Existence of non-isomorphic graphs that are quantum isomorphic.

Abstract

We introduce a two-player nonlocal game, called the $(G,H)$-isomorphism game, where classical players can win with certainty if and only if the graphs $G$ and $H$ are isomorphic. We then define the notions of quantum and non-signalling isomorphism, by considering perfect quantum and non-signalling strategies for the $(G,H)$-isomorphism game, respectively. In the quantum case, we consider both the tensor product and commuting frameworks for nonlocal games. We prove that non-signalling isomorphism coincides with the well-studied notion of fractional isomorphism, thus giving the latter an operational interpretation. Second, we show that, in the tensor product framework, quantum isomorphism is equivalent to the feasibility of two polynomial systems in non-commuting variables, obtained by relaxing the standard integer programming formulations for graph isomorphism to Hermitian variables. On the basis of this correspondence, we show that quantum isomorphic graphs are necessarily cospectral. Finally, we provide a construction for reducing linear binary constraint system games to isomorphism games. This allows us to produce quantum isomorphic graphs that are nevertheless not isomorphic. Furthermore, it allows us to show that our two notions of quantum isomorphism, from the tensor product and commuting frameworks, are in fact distinct relations, and that the latter is undecidable. Our construction is related to the FGLSS reduction from inapproximability literature, as well as the CFI construction.