no. 4
Continuous-time Quantum Walks on Cayley Graphs of Extraspecial Groups
Sin, Peter1 ; Sorci, Julien1
1 Department of Mathematics University of Florida P.O. Box 118105 Gainesville FL 32611, USA
Algebraic Combinatorics, Tome 5 (2022) no. 4, pp. 699-714.

We study continuous-time quantum walks on normal Cayley graphs of certain non-abelian groups called extraspecial groups. By applying general results for graphs in association schemes we determine the precise conditions for perfect state transfer and fractional revival. Using partial spreads, we construct Cayley graphs on extraspecial 2-groups that admit these phenomena. We also show that there is no normal Cayley graph of an extraspecial group that admits instantaneous uniform mixing.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/alco.237
Classification : 05C25, 81Q35
Mots clés : Cayley graph, perfect state transfer, fractional revival, uniform mixing, extraspecial groups
Sin, Peter 1 ; Sorci, Julien 1

1 Department of Mathematics University of Florida P.O. Box 118105 Gainesville FL 32611, USA 
@article{ALCO_2022__5_4_699_0,
     author = {Sin, Peter and Sorci, Julien},
     title = {Continuous-time {Quantum} {Walks} on {Cayley} {Graphs} of {Extraspecial} {Groups}},
     journal = {Algebraic Combinatorics},
     pages = {699--714},
     publisher = {The Combinatorics Consortium},
     volume = {5},
     number = {4},
     year = {2022},
     doi = {10.5802/alco.237},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/alco.237/}
}
TY  - JOUR
AU  - Sin, Peter
AU  - Sorci, Julien
TI  - Continuous-time Quantum Walks on Cayley Graphs of Extraspecial Groups
JO  - Algebraic Combinatorics
PY  - 2022
SP  - 699
EP  - 714
VL  - 5
IS  - 4
PB  - The Combinatorics Consortium
UR  - http://www.numdam.org/articles/10.5802/alco.237/
DO  - 10.5802/alco.237
LA  - en
ID  - ALCO_2022__5_4_699_0
ER  - 
%0 Journal Article
%A Sin, Peter
%A Sorci, Julien
%T Continuous-time Quantum Walks on Cayley Graphs of Extraspecial Groups
%J Algebraic Combinatorics
%D 2022
%P 699-714
%V 5
%N 4
%I The Combinatorics Consortium
%U http://www.numdam.org/articles/10.5802/alco.237/
%R 10.5802/alco.237
%G en
%F ALCO_2022__5_4_699_0
Sin, Peter; Sorci, Julien. Continuous-time Quantum Walks on Cayley Graphs of Extraspecial Groups. Algebraic Combinatorics, Tome 5 (2022) no. 4, pp. 699-714. doi : 10.5802/alco.237. http://www.numdam.org/articles/10.5802/alco.237/

[1] Bašić, Milan Characterization of circulant networks having perfect state transfer, Quantum Inf. Process., Volume 12 (2013) no. 1, pp. 345-364 | DOI | MR | Zbl

[2] Bernasconi, Anna; Godsil, Chris; Severini, Simone Quantum networks on cubelike graphs, Phys. Rev. A, Volume 78 (2008) no. 5, 052320, 5 pages | MR

[3] Cao, Xiwang; Chen, Bocong; Ling, San Perfect state transfer on Cayley graphs over dihedral groups, the non-normal case, Electron. J. Combin., Volume 27 (2020) no. 2, 2.28, 18 pages | MR | Zbl

[4] Cao, Xiwang; Feng, Keqin Perfect state transfer on Cayley graphs over dihedral groups, Linear Multilinear Algebra., Volume 69 (2021) no. 2, pp. 343-360 | MR | Zbl

[5] Cao, Xiwang; Wan, Jinlong; Feng, Keqin Some results on uniform mixing on abelian Cayley graphs (2019) | arXiv

[6] Chan, Ada Complex Hadamard matrices, instantaneous uniform mixing and cubes, Algebr. Comb., Volume 3 (2020) no. 3, pp. 757-774 | MR | Zbl

[7] Chan, Ada; Coutinho, Gabriel; Tamon, Christino; Vinet, Luc; Zhan, Hanmeng Fractional revival and association schemes, Discrete Math., Volume 343 (2020) no. 11, 112018, 15 pages | MR | Zbl

[8] Chan, Ada; Johnson, Bobae; Liu, Mengzhen; Schmidt, Malena; Yin, Zhanghan; Zhan, Hanmeng Laplacian fractional revival on graphs, Electron. J. Comb., Volume 28 (2021) no. 3, 3.22, 28 pages | MR | Zbl

[9] Cheung, Wang-Chi; Godsil, Chris Perfect state transfer in cubelike graphs, Linear Algebra Appl., Volume 435 (2011), pp. 2468-2474 | DOI | MR | Zbl

[10] Coutinho, Gabriel Quantum state transfer in graphs, Ph. D. Thesis, University of Waterloo (2014)

[11] Coutinho, Gabriel; Godsil, Chris; Guo, Krystal; Vanhove, Frédéric Perfect state transfer on distance regular graphs and association schemes, Linear Algebra Appl., Volume 478 (2015), pp. 108-130 | DOI | MR | Zbl

[12] Delsarte, Philippe An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl. (1973) no. 10, p. vi+97 | MR | Zbl

[13] Gerhardt, Heath; Watrous, John Continuous-Time Quantum Walks on the Symmetric Group, Approximation, randomization, and combinatorial optimization (Lecture Notes in Comput. Sci.), Volume 2764, Springer, Berlin, 2003, pp. 290-301 | DOI | MR | Zbl

[14] Godsil, Chris Periodic graphs, Electron. J. Comb., Volume 18 (2011) no. 1, 23, 15 pages | MR | Zbl

[15] Godsil, Chris; Mullin, Natalie; Roy, Aidan Uniform mixing and association schemes, Electron. J. Comb., Volume 24 (2017) no. 3, 3.22, 25 pages | MR | Zbl

[16] Gorenstein, Daniel Finite Groups, Chelsea Publishing Co., New York, 1980, xvii+519 pages

[17] Mortimer, Brian The modular permutation representations of the known doubly transitive groups, Proc. London Math. Soc. (3), Volume 41 (1980) no. 1, pp. 1-20 | DOI | MR | Zbl

[18] Tan, Ying-Ying; Feng, Keqin; Cao, Xiwang Perfect state transfer on abelian Cayley graphs, Linear Algebra Appl., Volume 563 (2019), pp. 331-352 | MR | Zbl

Cité par Sources :