Discrete-time Quantum Walk on the Cayley Graph of the Dihedral Group

TL;DR

This paper develops a spectral analysis of discrete-time quantum walks on the Cayley graph of the dihedral group, revealing connections between quantum walks with and without memory through theoretical and numerical analysis.

Contribution

It introduces a novel quantum walk model on the dihedral group's Cayley graph using Fourier analysis, linking memoryless and memory-based quantum walks.

Findings

Spectral properties of the dihedral quantum walk are characterized.

Quantum walk with memory relates to the walk without memory on a cycle.

Numerical simulations confirm theoretical predictions.

Abstract

The finite dihedral group generated by one rotation and one flip 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 conduct the model of discrete-time quantum walk on the Cayley graph of the dihedral group by special coding mode. This construction makes Fourier transformation can be used to carry out spectral analysis of the dihedral quantum walk, i.e. the non-abelian case. Furthermore, the relation between quantum walk without memory on the Cayley graph of the dihedral group and quantum walk with memory on a cycle is discussed, so that we can explore the potential of quantum walks without and with memory. Here, the numerical simulation is carried out to verify the theoretical analysis results and other properties of the proposed model are further studied.