Factoring Discrete Quantum Walks on Distance Regular Graphs into Continuous Quantum Walks

TL;DR

This paper demonstrates how discrete Grover quantum walks on distance regular graphs can be decomposed into products of continuous quantum walks, revealing a structural connection between discrete and continuous quantum dynamics on these graphs.

Contribution

It provides a novel factorization of the Grover walk transition matrix into continuous quantum walk components on distance digraphs, extending to graphs in Bose Mesner algebra.

Findings

Transition matrix squared factors into continuous walk products

Applicable to distance regular graphs with invertible adjacency matrices

Extends to graphs in Bose Mesner algebra

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.