Uniform Mixing in Chiral Quantum Walks

TL;DR

This paper investigates uniform mixing in chiral quantum walks, demonstrating new cases of uniform mixing in complete graphs with unitary signing and faster mixing in certain Hamming graphs, challenging previous no-go results.

Contribution

It introduces a stopping rule technique for quantum walks that achieves uniform mixing in new graph classes, including oriented circulants and specific Hamming graphs.

Findings

Complete graphs with unitary signing can have probabilistic uniform mixing.

An orientation of Hamming graphs mixes faster to uniform than previously known.

Infinite families of oriented circulants exhibit average uniform mixing.

Abstract

This paper studies uniform mixing in continuous-time quantum walks. We show that for some unitary signing $\sigma$, the complete graph $K^\sigma_n$ has probabilistic uniform mixing. In contrast, Ahmadi \etal (2003) proved that no complete graph has uniform mixing except for $K_2$, $K_3$, and $K_4$. Our technique is based on a stopping rule for quantum walks which reduces global to local uniform mixing. As a corollary, we found an orientation of $H(n,4)$ that mixes to uniform faster than any other Hamming graphs, which improves a result of Godsil and Zhan (2019). We also show that there are infinite families of oriented circulants with average uniform mixing. This is a chiral violation of a No-Go theorem due to Godsil (2013) which states that no graph has average uniform mixing except for $K_2$.