Uniform Mixing in Chiral Quantum Walks

Abstract: This paper studies uniform mixing in continuous-time quantum walks. We show that for some unitary signing $σ$ the complete graph $K^σ_n$ has probabilistic uniform mixing. In contrast, it is known {\em 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 special case, we found an orientation of $H(n,4)$ that mixes to uniform faster than any other Hamming graphs. We also show that there are infinite families of oriented circulants with {\em average} uniform mixing. This is a chiral violation of Godsil's {\em No-Go} theorem which states that no graph has average uniform mixing except for $K_2$.