Uniform mixing on Cayley graphs (Q2401400)

Summary: 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 \(2\pi/9\). Our second result shows that for every integer \(k\geq 3\), we can construct Cayley graphs over \(\mathbb{Z}_q^d\) that admit uniform mixing at time \(2\pi/q^k\), 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.