Uniform mixing in continuous-time quantum walks on oriented, nonabelian Cayley graphs

TL;DR

The paper identifies an infinite family of oriented, normal Cayley graphs on Suzuki $2$-groups $A(n,\theta)$ that exhibit uniform mixing for continuous-time quantum walks. It develops a general spectral-character framework for UM on oriented Cayley graphs and applies it to a carefully chosen central difference-set-based connection set $C$, proving UM at time $\tau=\pi/2^{n+1}$. The construction relies on a complete description of the irreducible characters of $A(n,\theta)$, which consist of $2^n$ linear characters and $2(2^n-1)$ nonlinear characters of degree $2^m$, enabling explicit eigenvalue calculations. This work links UM phenomena to central difference sets and Hadamard-like structures in nonabelian groups, broadening the catalog of UM examples beyond small graphs and contributing tools for analyzing quantum walks on oriented graphs.

Abstract

A family of oriented, normal, nonabelian Cayley graphs is presented, whose continuous-time quantum walks exhibit uniform mixing.

Theorems & Definitions (3)

• Lemma 3.1
• proof
• Theorem 3.2