Symmetries in discrete time quantum walks on Cayley graphs

TL;DR

This paper investigates the symmetries of discrete time quantum walks on Cayley graphs, providing a comprehensive theoretical framework and a constructive method to identify all symmetries, demonstrated through an example on a line.

Contribution

It introduces a rigorous formulation of quantum walk symmetries and offers a constructive approach to find all solutions with minimal assumptions.

Findings

Complete characterization of symmetries for quantum walks on Cayley graphs

Constructive method to determine all symmetry solutions

Application to quantum walk on a line demonstrating practical relevance

Abstract

We address the question of symmetries of an important type of quantum walks. We introduce all the necessary definitions and provide a rigorous formulation of the problem. Using a thorough analysis, we reach the complete answer by presenting a constructive method of finding all solutions of the problem with minimal additional assumptions. We apply the results on an example of a quantum walk on a line to demonstrate the practical significance of the theory.