mixing: can a quantum walk beginning at a single vertex result in a state whose probability distribution is uniform over all vertices? If this is possible at some small time $t$, then the behaviour is quite different from a classical walk, which typically approaches the uniform distribution as $t \to \infty$.

Moore and Russell [@moore02] in 2001 gave the first example of uniform mixing, showing that the $n$-cube has uniform mixing at time $t = n\pi/4$. Since then a large body of work has been produced to study this problem. We summarize the conclusions. The following graphs do admit uniform mixing:

(a) The complete graphs $K_2$, $K_3$, and $K_4$.

(b) The Hamming graphs $H(n,q)$, $q \in \{2,3,4\}$ (see [@moore02] [@carlson07]).

(c) The Cartesian product of any two graphs that admit uniform mixing at the same time.

Note that since the Hamming graph $H(n,q)$ is the $n$-th Cartesian power of $K_q$, the results in (b) follow from the results in (a). The Cartesian powers of $K_2$ and $K_4$ admit uniform mixing when $t=\pi/4$, hence the products of powers $K_2^{{\mathbin{\square}}r}\times K_4^{{\mathbin{\square}}s}$) also admit uniform mixing. A number of Cayley graphs for ${{\mathbb Z}}_2^d$ admit uniform mixing, but no classification is yet known. 