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arXiv · 2605.02254

Perfect state transfer in Grover walks on dihedral Cayley graphs

Koushik Bhakta·Bikash Bhattacharjya·Xiwang Cao
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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.

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Koushik Bhakta, Bikash Bhattacharjya, Xiwang Cao. 2026-05-04. Perfect state transfer in Grover walks on dihedral Cayley graphs. https://arxiv.org/abs/2605.02254

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