Perfect state transfer in Grover walks on dihedral Cayley graphs

TL;DR

This paper characterizes when perfect state transfer occurs in Grover walks on Cayley graphs over dihedral groups, revealing conditions based on group properties and graph normality.

Contribution

It provides a complete characterization of PST in Cayley graphs over dihedral groups, including necessary and sufficient conditions for various cases.

Findings

PST does not occur for normal Cayley graphs when n is odd.

The paper constructs infinite families of Cayley graphs with PST.

Representation theory of dihedral groups underpins the analysis.

Abstract

The paper investigates perfect state transfer (PST) in Grover walks on Cayley graphs over the dihedral group $D_n$. The Grover walk is a discrete-time quantum walk widely studied in quantum information processing. A Cayley graph $\operatorname{Cay}(\Gamma,S)$ is called normal if $S$ is the union of some conjugacy classes of the group $\Gamma$; otherwise, it is called non-normal. Most existing studies have been restricted to Cayley graphs over abelian groups. In contrast, we investigate both normal and non-normal cases for Cayley graphs over the non-abelian group $D_n$. By examining the parity of $n$ and the normality of the Cayley graph, we obtain a complete characterization of PST on $\operatorname{Cay}(D_n,S)$. In particular, we establish necessary and sufficient conditions for the occurrence of PST in all possible cases, and prove that PST does not occur for normal Cayley graphs when $n$ is odd. Furthermore, we construct several infinite families of normal and non-normal Cayley graphs $\operatorname{Cay}(D_n,S)$ that exhibit PST, illustrating the application of the main result. Our approach is based on the representation theory of the dihedral group.