$ε$-Uniform Mixing in Discrete Quantum Walks

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.