Discrete-time quantum walks on Cayley graphs of Dihedral groups using generalized Grover coins

TL;DR

This paper investigates discrete-time quantum walks on Cayley graphs of Dihedral groups using generalized Grover coins, revealing conditions for periodicity and demonstrating localization phenomena through numerical simulations.

Contribution

It introduces a family of coins as linear combinations of permutation matrices and analyzes their effects on quantum walk properties, including periodicity and localization.

Findings

Quantum walks are periodic only for permutation or negative permutation coins.

Localization occurs for a wide range of coins across different graph sizes.

Numerical simulations confirm localization phenomena in the studied quantum walks.

Abstract

In this paper we study discrete-time quantum walks on Cayley graphs corresponding to Dihedral groups, which are graphs with both directed and undirected edges. We consider the walks with coins that are one-parameter continuous deformation of the Grover matrix and can be written as linear combinations of certain permutation matrices. We show that the walks are periodic only for coins that are permutation or negative of a permutation matrix. Finally, we investigate the localization property of the walks through numerical simulations and observe that the walks localize for a wide range of coins for different sizes of the graphs.