We study whether a discrete quantum walk can get arbitrarily close to a state whose Schur square is constant on all 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 at every vertex, and the states we get arbitrarily close to are constant on the outgoing arcs of the vertices, then 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.

翻译：我们研究离散量子行走能否在从某一顶点的出弧均匀叠加态出发时，任意接近其Schur平方在所有弧上为常数的状态。利用图的邻接谱，我们刻画了非二部图上这一现象的特性；特别地，若该现象在每个顶点处发生，且我们任意接近的状态在顶点出弧上为常数，则图的邻接代数包含一个实（正则）Hadamard矩阵。随后，我们发现了允许该现象存在的无限族本原强正则图。