Complex Hadamard Matrices, Instantaneous Uniform Mixing and Cubes

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We study the continuous-time quantum walks on graphs in the adjacency algebra of the $n$-cube and its related distance regular graphs.

• For $k\geq 2$, we find graphs in the adjacency algebra of $(2^{k+2}-8)$-cube that admit instantaneous uniform mixing at time $\pi/2^k$ and graphs that have perfect state transfer at time $\pi/2^k$.
• 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.