# ALGEBRAIC COMBINATORICS

Complex Hadamard matrices, instantaneous uniform mixing and cubes
Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 757-774.

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

For $k\ge 2$, we find graphs in the adjacency algebra of $\left({2}^{k+2}-8\right)$-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.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.112
Classification: 05E03
Keywords: Association schemes, Hamming schemes, complex Hadamard matrix, continuous-time quantum walks, instantaneous uniform mixing, perfect state transfer.

1 York University Dept. of Mathematics and Statistics 4700 Keele Street Toronto Ontario M3J 1P3, Canada
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Chan, Ada. Complex Hadamard matrices, instantaneous uniform mixing and cubes. Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 757-774. doi : 10.5802/alco.112. https://alco.centre-mersenne.org/articles/10.5802/alco.112/

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