Paley graph of order 9 and certain graphs corresponding to regular symmetric Hadamard matrices with constant diagonal.

Since uniform mixing seems to be rare, we consider a relaxed version, called $\epsilon$-uniform mixing, and we demonstrate that this does take place on all cycles of prime length.

Uniform mixing
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Let $A$ denote the adjacency matrix of a graph $X$ on $n$ vertices. The transition operator $U(t)$ defined by $$U(t) = e^{itA}.$$ determines a continuous quantum walk. It is straightforward to observe that the transition matrix satisfies the following properties.

(i) [$U(t)^{T} = U(t).$]{}

(ii) [$\overline{U(t)} = U(-t).$]{}

(iii) [$U(t)$ is unitary.]{}

We say the graph $X$ admits *uniform mixing* at time $t$ if and only if $$U(t) \circ U(-t) = \frac{1}{n}J$$ where $J$ is the $n \times n$ all-ones matrix.

We say that $\epsilon$-uniform mixing occurs if, for each $\epsilon > 0$, there is a time $t$ such that $$||U(t) \circ U(-t) - \frac{1}{n} J || < \epsilon.$$

It is useful to note the following lemma, which relates uniform mixing on a graph and its complement at certain times.

Let $X$ denote a regular graph on $n$ vertices, and let $t$ denote an integer multiple of $2 \pi/n$. At time 