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}, publisher = {MathOA foundation}, volume = {3}, number = {3}, year = {2020}, pages = {757-774}, 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/

[1] On mixing in continuous-time quantum walks on some circulant graphs, Quantum Inf. Comput., Volume 3 (2003) no. 6, pp. 611-618 | MR 2023606

[2] Mixing of Quantum Walks on Generalized Hypercubes, International Journal of Quantum Information, Volume 6 (2008) no. 6, pp. 1135-1148 | Article | Zbl 1165.81309

[3] Distance-regular graphs, Springer-Verlag, Berlin, 1989, xviii+495 pages | MR 1002568 | Zbl 0747.05073

[4] On the residues of binomial coefficients and their products modulo prime powers, Acta Math. Sin. (Engl. Ser.), Volume 18 (2002) no. 2, pp. 277-288 | Article | MR 1910963 | Zbl 1026.11005

[5] Perfect state transfer in cubelike graphs, Linear Algebra Appl., Volume 435 (2011) no. 10, pp. 2468-2474 | Article | MR 2811131 | Zbl 1222.05150

[6] Zeros of generalized Krawtchouk polynomials, J. Approx. Theory, Volume 60 (1990) no. 1, pp. 43-57 | Article | MR 1028893 | Zbl 0693.33005

[7] Universal computation by quantum walk, Phys. Rev. Lett., Volume 102 (2009) no. 18, 180501, 4 pages | Article | MR 2507892

[8] Perfect transfer of arbitrary states in quantum spin networks, Phys. Rev. A, Volume 71 (2005) no. 3, 032312, 11 pages | Article

[9] History of the theory of numbers. Vol. I: Divisibility and primality, Chelsea Publishing Co., New York, 1966, xii+486 pages | MR 0245499 | Zbl 0958.11500

[10] Quantum computation and decision trees, Phys. Rev. A (3), Volume 58 (1998) no. 2, pp. 915-928 | Article | MR 1638221

[11] Generalized Hamming Schemes (2010) (https://arxiv.org/abs/1011.1044)

[12] State transfer on graphs, Discrete Math., Volume 312 (2012) no. 1, pp. 129-147 | Article | MR 2852516 | Zbl 1232.05123

[13] Quantum random walks: an introductory overview, Contemporary Physics, Volume 44 (2003) no. 4, pp. 307-327 | Article

[14] Quantum walks on the hypercube, Randomization and approximation techniques in computer science (Lecture Notes in Comput. Sci.) Volume 2483, Springer, Berlin, 2002, pp. 164-178 | Article | MR 2047028 | Zbl 1028.68570