Uniform Mixing on Cayley Graphs

TL;DR

This paper characterizes specific Cayley graphs over finite groups that exhibit uniform quantum mixing, introduces new constructions for such graphs, and identifies the first irregular graphs with this property.

Contribution

It provides a complete characterization of certain regular Cayley graphs over
7d_3^d and constructs new Cayley graphs over
7d_q^d with uniform mixing at specified times.

Findings

Characterization of all 2(d+2)-regular Cayley graphs over
7d_3^d with uniform mixing at 2c/9

Construction of Cayley graphs over
7d_q^d with uniform mixing at 2c/q^k for q=3,4

Identification of the first irregular graphs, Cartesian powers of K_{1,3}, with uniform mixing

Abstract

We provide new examples of Cayley graphs on which the quantum walks reach uniform mixing. Our first result is a complete characterization of all $2(d+2)$-regular Cayley graphs over $\mathbb{Z}_3^d$ that admit uniform mixing at time $2\pi/9$. Our second result shows that for every integer $k\ge 3$, we can construct Cayley graphs over $\mathbb{Z}_q^d$ that admit uniform mixing at time $2\pi/q^k$, where $q=3, 4$. We also find the first family of irregular graphs, the Cartesian powers of the star $K_{1,3}$, that admit uniform mixing.