Three-state quantum walk on the Cayley Graph of the Dihedral Group

TL;DR

This paper introduces a three-state discrete-time quantum walk model on the Cayley graph of the dihedral group, deriving analytical results and exploring localization effects, which can inform quantum algorithm development.

Contribution

It presents a novel three-state quantum walk model on the dihedral group's Cayley graph with analytical expressions and numerical analysis of localization phenomena.

Findings

Localization depends on group size, coin operator, and initial state

Analytical expressions for position distribution and return probability

Numerical simulations reveal rich quantum walk behaviors

Abstract

The finite dihedral group generated by one rotation and one reflection is the simplest case of the non-abelian group. Cayley graphs are diagrammatic counterparts of groups. In this paper, much attention is given to the Cayley graph of the dihedral group. Considering the characteristics of the elements in the dihedral group, we propose a model of three-state discrete-time quantum walk (DTQW) on the Caylay graph of the dihedral group with Grover coin. We derive analytic expressions for the the position probability distribution and the long-time limit of the return probability starting from the origin. It is shown that the localization effect is governed by the size of the underlying dihedral group, coin operator and initial state. We also numerically investigate the properties of the proposed model via the probability distribution and the time-averaged probability at the designated position. The abundant phenomena of three-state Grover DTQW on the Caylay graph of the dihedral group can help the community to better understand and to develop new quantum algorithms.