Complex Hadamard matrices, instantaneous uniform mixing and cubes

Abstract

We study the continuous-time quantum walks on graphs in the adjacency algebra of the n-cube and its related distance regular graphs. For k > 2, we find graphs in the adjacency algebra of (2k+2 − 8)-cube that admit instantaneous uniform mixing at time π/2k and graphs that have perfect state transfer at time π/2k. We characterize the folded n-cubes, the halved n-cubes and the folded halved n-cubes whose adjacency algebra contains a complex Hadamard matrix. We obtain the same conditions for the characterization of these graphs admitting instantaneous uniform mixing.