Two-particle quantum walks applied to the graph isomorphism problem

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.