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Fractional revival on Cayley graphs over abelian groups

Jing Wang, Ligong Wang, Xiaogang Liu

TL;DR

This work determines when fractional revival occurs on Cayley graphs over finite abelian groups, unifying FR criteria across broad graph families. It derives a complete necessary-and-sufficient condition expressed in group-theoretic and spectral terms: FR between $v$ and $v+a$ requires $a$ to have order two, evenness of each $n_s$ where $a_s\neq0$, and a time condition $\frac{t}{2\pi}(\lambda_x-\lambda_y)\in \mathbb{Z}$ for all $(x,y)$ in a set $N$ defined by the order relation and a parity constraint, with eigenvalues $\lambda_x=\sum_{g\in S}\prod_{s=1}^r e^{2\pi i x_s g_s / n_s}$. The approach uses Fourier analysis on the group, spectral decomposition, and character theory to reduce FR to arithmetic on eigenvalue differences, yielding minimum FR times $t=\frac{2\pi}{M}$ where $M$ is a gcd of those differences. The results specialize to circulant graphs ($r=1$) and cubelike graphs ($n_s=2$), with concrete examples and corollaries that clarify when FR occurs and how to compute the corresponding FR times. These findings provide a robust framework for designing quantum networks with FR on abelian-group based structures.

Abstract

In this paper, we investigate the existence of fractional revival on Cayley graphs over finite abelian groups. We give a necessary and sufficient condition for Cayley graphs over finite abelian groups to have fractional revival. As applications, the existence of fractional revival on circulant graphs and cubelike graphs are characterized.

Fractional revival on Cayley graphs over abelian groups

TL;DR

This work determines when fractional revival occurs on Cayley graphs over finite abelian groups, unifying FR criteria across broad graph families. It derives a complete necessary-and-sufficient condition expressed in group-theoretic and spectral terms: FR between and requires to have order two, evenness of each where , and a time condition for all in a set defined by the order relation and a parity constraint, with eigenvalues . The approach uses Fourier analysis on the group, spectral decomposition, and character theory to reduce FR to arithmetic on eigenvalue differences, yielding minimum FR times where is a gcd of those differences. The results specialize to circulant graphs () and cubelike graphs (), with concrete examples and corollaries that clarify when FR occurs and how to compute the corresponding FR times. These findings provide a robust framework for designing quantum networks with FR on abelian-group based structures.

Abstract

In this paper, we investigate the existence of fractional revival on Cayley graphs over finite abelian groups. We give a necessary and sufficient condition for Cayley graphs over finite abelian groups to have fractional revival. As applications, the existence of fractional revival on circulant graphs and cubelike graphs are characterized.
Paper Structure (2 sections, 6 theorems, 50 equations, 4 figures)

This paper contains 2 sections, 6 theorems, 50 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['FRCAY-mian result']}

Key Result

Theorem 1.1

Let $\Gamma=\mathbb{Z}_{n_1}\oplus \mathbb{Z}_{n_2}\oplus\cdots \oplus \mathbb{Z}_{n_r}$ be a finite abelian group of order $|\Gamma|=n_1n_2\cdots n_r$ as shown in (GammaDef-1), and $S$ a subset of $\Gamma$ such that $0\notin S$ and $-S=S$. Let $n=(n_1,n_2,\ldots, n_r)$ and Then $\mathrm{Cay}(\Gamma, S)$ has FR at time $t$ between vertices $v$ and $v+a$ if and only if the following three conditio

Figures (4)

  • Figure 1: $\mathrm{Cay}(\mathbb{Z}_6, S_1)$
  • Figure 2: $\mathrm{Cay}(\mathbb{Z}_4, S_2)$
  • Figure 3: $\mathrm{Cay}(\mathbb{Z}_6, S_3)$
  • Figure 4: $\mathrm{Cay}(\mathbb{Z}_2\oplus \mathbb{Z}_2\oplus\mathbb{Z}_2, S_4)$

Theorems & Definitions (12)

  • Theorem 1.1
  • Corollary 2.1
  • Example 1
  • Example 2
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Example 3
  • Corollary 2.4
  • Example 4
  • ...and 2 more