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 $({2}^{k+2}-8)$-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.

Revised: 2020-02-09

Accepted: 2020-02-09

Published online: 2020-06-02

Classification: 05E03

Keywords: Association schemes, Hamming schemes, complex Hadamard matrix, continuous-time quantum walks, instantaneous uniform mixing, perfect state transfer.

@article{ALCO_2020__3_3_757_0, author = {Chan, Ada}, title = {Complex Hadamard matrices, instantaneous uniform mixing and cubes}, journal = {Algebraic Combinatorics}, pages = {757--774}, publisher = {MathOA foundation}, volume = {3}, number = {3}, year = {2020}, doi = {10.5802/alco.112}, language = {en}, url = {alco.centre-mersenne.org/item/ALCO_2020__3_3_757_0/} }

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/item/ALCO_2020__3_3_757_0/

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