We consider a discrete-time quantum walk, called the Grover walk, on a distance regular graph
X. Given that
X has diameter
d and invertible adjacency matrix, we show that the square of the transition matrix of the Grover walk on
X is a product of at most
d commuting transition matrices of continuous-time quantum walks, each on some distance digraph of the line digraph of
X. We also obtain a similar factorization for any graph
X in a Bose Mesner algebra.
Various quantum walk-based algorithms have been developed, aiming to distinguish non-isomorphic graphs with polynomial scaling, within both the discrete-time quantum walk (DTQW) and continuous-time quantum walk (CTQW) frameworks. Whilst both the single-particle DTQW and CTQW have failed to distinguish non-isomorphic strongly regular graph families (prompting the move to multi-particle graph isomorphism (GI) algorithms), the single-particle DTQW has been successfully modified by the introduction of a phase factor to distinguish a wide range of graphs in polynomial time. In this paper, we prove that an analogous phase modification to the single particle CTQW does not have the same distinguishing power as its discrete-time counterpart, in particular it cannot distinguish strongly regular graphs with the same family parameters with the same efficiency.