Uniform mixing on integral abelian Cayley graph

In the past few decades, quantum algorithms have become a popular research area of both mathematicians and engineers. Among them, uniform mixing provides a uniform probability distribution of quantum information over time which attracts a special attention. However, there are only a few known examples of graphs which admit uniform mixing. In this paper, a characterization of abelian Cayley graphs having uniform mixing is presented. Some concrete constructions of such graphs are provided. Specifically, for cubelike graphs, it is shown that the Cayley graph ${\rm Cay}(\mathbb{F}_2^{2k};S)$ has uniform mixing if the characteristic function of $S$ is bent. Moreover, a difference-balanced property of the eigenvalues of an integral abelian Cayley graph having uniform mixing is established. Some nonexistence results of uniform mixing on abelian Cayley graphs are presented also. Notably, for a linear abelian Cayley graph $\Gamma$ over $\mathbb{Z}_n^r$, it is proved that uniform mixing occurs on this graph only if $n=2,3,4$ which partially confirms a long-standing conjecture. Finally, with some restrictions on the connection set, we show that if a cyclic group $G$ whose order is divisible by an odd prime such that ${\rm Cay}(G,S)$ has uniform mixing at some time $t$ being a rational multiple of $\pi$, then the graph should be integral. An interesting reduction method on the circulant Cayley graphs having uniform mixing has been established. Using this method, we can answer some related open questions. For example, we can show that the only two cycles having uniform mixing are $C_3$ and $C_4$.