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We investigate the quantum dynamics of particles on graphs ("quantum randomwalks"), with the aim of developing quantum algorithms for determining if twographs are isomorphic (related to each other by a relabeling of vertices). Wefocus on quantum random walks of multiple non-interacting particles on stronglyregular graphs (SRGs), a class of graphs with high symmetry that is known tohave pairs of graphs that are hard to distinguish. Previous work has alreadydemonstrated analytically that two-particle non-interacting quantum walkscannot distinguish non-isomorphic SRGs of the same family. Here, we demonstratenumerically that three-particle non-interacting quantum walks have significant,but not universal, distinguishing power for pairs of SRGs, proving afundamental difference between the distinguishing power of two-particle andthree-particle non-interacting walks. We analytically show why thisdistinguishing power is possible, whereas it is forbidden for two-particlenon-interacting walks. Based on sampling of SRGs with up to 64 vertices, wefind no difference in the distinguishing power of bosonic and fermionic walks.In addition, we find that the four-fermion non-interacting walk has greaterdistinguishing power than the three-particle walks on SRGs, showing thatincreasing particle number increases distinguishing power. However, we alsoanalytically show that no non-interacting walk with a fixed number of particlescan distinguish all SRGs, thus demonstrating a potential fundamental differencebetween the distinguishing power of interacting and noninteracting walks.

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