We study whether a discrete quantum walk can get arbitrarily close to a state whose entries have the same absolute value over all the arcs, given that the walk starts with a uniform superposition of the outgoing arcs of some vertex. We characterize this phenomenon on non-bipartite graphs using the adjacency spectrum of the graph; in particular, if this happens in some association scheme and the state we get arbitrarily close to ``respects the neighborhood", then it happens regardless of the initial vertex, and the adjacency algebra of the graph contains a real (regular) Hadamard matrix. We then find infinite families of primitive strongly regular graphs that admit this phenomenon. We also derive some results on a strengthening of this phenomenon called simultaneous $\epsilon$-uniform mixing, which enables local $\epsilon$-uniform mixing at every vertex.

翻译：我们研究离散量子游走能否从某个顶点的出弧均匀叠加态出发，无限接近所有弧上分量模长均相等的状态。利用图的邻接谱刻画了非二部图上的这一现象；特别地，若该现象发生于某个结合方案且无限接近的状态“尊重邻域”，则其发生与初始顶点无关，且图的邻接代数中包含一个实（正则）阿达马矩阵。进而发现了允许该现象的无限族本原强正则图。此外，我们推导了关于该现象的强化形式——同步$ε$-均匀混合的一些结果，该强化形式可实现每个顶点处的局部$ε$-均匀混合。