  Classical random walks on well-behaved graphs are rapidly mixing towards the
uniform distribution. Moore and Russell showed that a continuous quantum walk
on the hypercube is instantaneously uniform mixing. We show that the
continuous-time quantum walks on other well-behaved graphs do not exhibit this
uniform mixing. We prove that the only graphs amongst balanced complete
multipartite graphs that have the instantaneous uniform mixing property are the
complete graphs on two, three and four vertices, and the cycle graph on four
vertices. Our proof exploits the circulant structure of these graphs.
Furthermore, we conjecture that most complete cycles and Cayley graphs lack
this mixing property as well.
