Two-particle quantum walks applied to the graph isomorphism problem

TL;DR

This paper demonstrates that interacting two-particle quantum walks can distinguish non-isomorphic strongly regular graphs, outperforming noninteracting particles, and provides both numerical and analytical evidence for this advantage.

Contribution

It shows that interacting two-particle quantum walks can solve the graph isomorphism problem for strongly regular graphs, a task noninteracting walks cannot accomplish.

Findings

Interacting two-particle quantum walks distinguish all examined non-isomorphic SRGs.

Noninteracting quantum walks cannot distinguish certain pairs of SRGs.

Numerical analysis involved over 500 million comparisons of evolution operators.

Abstract

We show that the quantum dynamics of interacting and noninteracting quantum particles are fundamentally different in the context of solving a particular computational problem. Specifically, we consider the graph isomorphism problem, in which one wishes to determine whether two graphs are isomorphic (related to each other by a relabeling of the graph vertices), and focus on a class of graphs with particularly high symmetry called strongly regular graphs (SRG's). We study the Green's functions that characterize the dynamical evolution single-particle and two-particle quantum walks on pairs of non-isomorphic SRG's and show that interacting particles can distinguish non-isomorphic graphs that noninteracting particles cannot. We obtain the following specific results: (1) We prove that quantum walks of two noninteracting particles, Fermions or Bosons, cannot distinguish certain pairs of non-isomorphic SRG's. (2) We demonstrate numerically that two interacting Bosons are more powerful than single particles and two noninteracting particles, in that quantum walks of interacting bosons distinguish all non-isomorphic pairs of SRGs that we examined. By utilizing high-throughput computing to perform over 500 million direct comparisons between evolution operators, we checked all tabulated pairs of non-isomorphic SRGs, including graphs with up to 64 vertices. (3) By performing a short-time expansion of the evolution operator, we derive distinguishing operators that provide analytic insight into the power of the interacting two-particle quantum walk.