Factoring discrete-time quantum walks on distance regular graphs into continuous-time quantum walks

Abstract

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.