Uniform mixing and $ε$-uniform mixing on cycles

Xiwang Cao; Cuiwen Zhu

Uniform mixing and $ε$-uniform mixing on cycles

Combinatorics 2026-07-05 v1 Quantum Physics

Authors: Xiwang Cao , Cuiwen Zhu

Abstract

We study continuous-time quantum walks on cycles. We prove two complementary results. Firstly, the cycle $C_9$ does not admit uniform mixing at any time. Using the similar idea and Dickson polynomials, we prove that $C_{15}$ does not admit uniform mixing at any time neither. Secondly, for every prime $p$ , we show that the cycle $C_{p^2}$ admits $\epsilon$ -uniform mixing.

Keywords

quantum walk uniform mixing cycle graph Dickson polynomial continuous-time

Cite

@article{arxiv.2607.04207,
  title  = {Uniform mixing and $ε$-uniform mixing on cycles},
  author = {Xiwang Cao and Cuiwen Zhu},
  journal= {arXiv preprint arXiv:2607.04207},
  year   = {2026}
}