Perfect state transfer on Cayley graphs over a non-abelian group of order $8n$

TL;DR

This paper investigates conditions under which Cayley graphs over a specific non-abelian group of order 8n exhibit perfect state transfer, contributing to quantum information transfer theory.

Contribution

It provides necessary and sufficient conditions for perfect state transfer on Cayley graphs over the non-abelian group V_{8n}.

Findings

Identifies conditions for PST on Cay(V_{8n}, S)

Characterizes PST in terms of group and subset properties

Advances understanding of quantum state transfer in non-abelian group structures

Abstract

The \textit{transition matrix} of a graph $\Gamma$ with adjacency matrix $A$ is defined by $H(\tau ) := \exp(-\mathbf{i}\tau A)$, where $\tau \in \mathbb{R}$ and $\mathbf{i} = \sqrt{-1}$. The graph $\Gamma$ exhibits \textit{perfect state transfer} (PST) between the vertices $u$ and $v$ if there exists $\tau_0(>0)\in \mathbb{R}$ such that $\lvert H(\tau_0)_{uv} \rvert = 1$. For a positive integer $n$, the group $V_{8n}$ is defined as $V_{8n} := \langle a,b \colon a^{2n} = b^{4} = 1, ba = a^{-1}b^{-1}, b^{-1}a = a^{-1}b \rangle$. In this paper, we study the existence of perfect state transfer on Cayley graphs $\text{Cay}(V_{8n}, S)$. We present some necessary and sufficient conditions for the existence of perfect state transfer on $\text{Cay}(V_{8n}, S)$.