Non-interacting multi-particle quantum random walks applied to the graph isomorphism problem for strongly regular graphs

We investigate the quantum dynamics of particles on graphs ("quantum random walks"), with the aim of developing quantum algorithms for determining if two graphs are isomorphic (related to each other by a relabeling of vertices). We focus on quantum random walks of multiple non-interacting particles on strongly regular graphs (SRGs), a class of graphs with high symmetry that is known to have pairs of graphs that are hard to distinguish. Previous work has already demonstrated analytically that two-particle non-interacting quantum walks cannot distinguish non-isomorphic SRGs of the same family. Here, we demonstrate numerically that three-particle non-interacting quantum walks have significant, but not universal, distinguishing power for pairs of SRGs, proving a fundamental difference between the distinguishing power of two-particle and three-particle non-interacting walks. We analytically show why this distinguishing power is possible, whereas it is forbidden for two-particle non-interacting walks. Based on sampling of SRGs with up to 64 vertices, we find no difference in the distinguishing power of bosonic and fermionic walks. In addition, we find that the four-fermion non-interacting walk has greater distinguishing power than the three-particle walks on SRGs, showing that increasing particle number increases distinguishing power. However, we also analytically show that no non-interacting walk with a fixed number of particles can distinguish all SRGs, thus demonstrating a potential fundamental difference between the distinguishing power of interacting and noninteracting walks.