$ε$-Uniform Mixing in Discrete Quantum Walks

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.