State Transfer on Unitary Cayley Graphs and Quadratic Unitary Cayley Graphs

TL;DR

This paper classifies when certain unitary Cayley graphs and quadratic unitary Cayley graphs exhibit quantum state transfer phenomena, including perfect and fractional revival, based on their structural properties.

Contribution

It provides a complete classification of state transfer properties in $X_n$ and $G_n$, revealing conditions for fractional revival, periodicity, and perfect state transfer.

Findings

$X_n$ admits fractional revival iff it admits pretty good fractional revival.

$G_n$ admits periodicity under specific conditions.

The paper characterizes all $G_n$ with perfect and pretty good state transfer.

Abstract

The unitary Cayley graph, denoted $X_n$, is the graph with vertex set ${\mathbb{Z}}_n$ such that two distinct vertices $a$ and $b$ are adjacent if $a-b=u$ for some $u$ with $1 \leq u \leq n-1$ and $\gcd(u,n) = 1$. The quadratic unitary Cayley graph, denoted $G_n$, is the graph with vertex set ${\mathbb{Z}}_n$ such that two distinct vertices $a$ and $b$ are adjacent if $a-b=u^2$ or $a-b=-u^2$ for some $u$ with $1 \leq u \leq n-1$ and $\gcd(u,n) = 1$. In this paper, we classify all $X_n$ admitting pretty good fractional. We also classify all $X_n$ that admit fractional revival. It turns out that $X_n$ admits fractional revival if and only if it admits pretty good fractional revival. Further, we classify all $G_n$ admitting periodicity. As a consequence, we obtain all $G_n$ admitting perfect state transfer. We also classify $G_n$ admitting pretty good state transfer, pretty good fractional revival and fractional revival.