Uniform Mixing on Cayley Graphs

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 $2\pi \mathrm{/}9$. 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 $2\pi \mathrm{/}{q}^k$, 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.