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.Comment: 12 pages, 3 figures, 3 table

Solving the graph-isomorphism problem with a quantum annealer

“…Gudkov and Nussinov [12] proposed a classical algorithm to distinguish non-isomorphic graphs by mapping them onto various physical problems. Shiau et al proved that the simplest classical algorithm fails to distinguish some pairs of nonisomorphic graphs and also proved that continuous-time one-particle quantum random walks cannot distinguish some non-isomorphic graphs [13][14][15]. More recently, it has been found that classical random walks and quantum random walks can exhibit qualitatively different properties [16][17][18].…”

“…There too, we find that it is sufficient to look at The largest size of graphs that we deal with analytically using perturbation theory is the family of 41 graphs having N = 29 vertices and signature (29,14,6,7). A scatterplot in the M x − Q 2 plane for this family is given in Fig.…”

“…The main results of this paper are for strongly regular graphs (SRGs), a class of graphs, subsets of which are known to be difficult to distinguish [14]. An SRG is a graph in which (i) all vertices have the same degree, (ii) each pair of neighboring vertices has the same number of shared neighbors, and (iii) each pair of non-neighboring vertices has the same number of shared neighbors.…”

A quantum-walk-inspired adiabatic algorithm for solving graph isomorphism problems

“…The standard benchmark for this approach is provided by the family of Strongly Regular Graphs (SRGs), that includes many hard instances of GI [6]. For example in [7][8][9] to distinguish non isomorphic graphs the authors exploit continuous [10][11][12] and discrete time quantum walks [13] of one or more particles moving through the graphs and compare the evolution of the same initial condition on the two graphs. The distinguishing power of the algorithm increases with the number of walker moving along the graph; the technique, however, is not universal and there are non-isomorphic graphs that cannot be distinguished.…”

Discrete-time interacting quantum walks and quantum Hash schemes

“…Therefore it is hoped that two-particle quantum walks are more powerful than single-particle quantum walks. Much attention has been paid to multi-particle quantum walks and a lot of correlative algorithms have been proposed [6,[8][9][10].…”