We study whether the probability distribution of a discrete quantum walk can get arbitrarily close to uniform, given that the walk starts with a uniform superposition of the outgoing arcs of some vertex. We establish a characterization of this phenomenon on regular non-bipartite graphs in terms of their adjacency eigenvalues and eigenprojections. Using theory from association schemes, we show this phenomenon happens on a strongly regular graph $X$ if and only if $X$ or $\overline{X}$ has parameters $(4m^2, 2m^2\pm m, m^2\pm m, m^2\pm m)$ where $m\ge 2$.

翻译：我们研究离散量子行走的概率分布能否无限趋近于均匀分布，前提是该行走始于某顶点出弧的均匀叠加态。我们在正则非二分图上基于邻接谱特征值和特征投影建立了该现象的完整刻画。利用结合方案理论，我们证明该现象在强正则图X上发生的充要条件是：X或其补图$\overline{X}$具有参数$(4m^2, 2m^2\pm m, m^2\pm m, m^2\pm m)$，其中$m\ge 2$。