Influence of coin symmetry on infinite hitting times in quantum walks

Classical random walks on finite graphs have an underrated property: a walk from any vertex can reach every other in time, provided they are connected. Discrete-time quantum connected however, infinite hitting times. This phenomenon is related to graph symmetry, as previously characterized by the group of direction-preserving automorphisms that trivially affect coin Hilbert space. If symmetric enough (in particular sense) then associated unitary will contain eigenvectors do not overlap set target vertices, for flip operator. These span Infinite Hitting Time (IHT) subspace. Quantum states IHT subspace never leading However, this whole story: 3D cube does satisfy symmetry constraint, yet with certain coins exhibit We study effect analyzing coin-permutation symmetries (CPS): act nontrivially space but leave operator invariant. Unitaries using highly large CPS groups, such permutation-invariant Grover coin, higher probabilities arriving, result their larger subspaces.