Browse issues
no. 1
Perfect state transfer in graphs related to linear groups in two dimensions
Pantangi, Venkata Raghu Tej1; Sin, Peter2
1Department of Mathematics and Statistics, University of Regina, Regina, SK S4S 0A2, Canada
2Department of Mathematics, University of Florida, P. O. Box 118105, Gainesville, FL 32611, USA
Algebraic Combinatorics, Volume 9 (2026) no. 1, pp. 261-287

We construct families of graphs from linear groups $\mathrm{SL}(2,q)$, $\mathrm{GL}(2,q)$ and $\mathrm{GU}(2,q)$, where $q$ is an odd prime power, with the property that the continuous-time quantum walks on the associated networks of qubits admit perfect state transfer.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.469
Classification: 05C25, 81Q35
Keywords: Cayley graph, perfect state transfer, linear groups in two dimensions

Pantangi, Venkata Raghu Tej  1 ; Sin, Peter  2

1 Department of Mathematics and Statistics, University of Regina, Regina, SK S4S 0A2, Canada
2 Department of Mathematics, University of Florida, P. O. Box 118105, Gainesville, FL 32611, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{ALCO_2026__9_1_261_0,
     author = {Pantangi, Venkata Raghu Tej and Sin, Peter},
     title = {Perfect state transfer in graphs related to linear groups in two dimensions},
     journal = {Algebraic Combinatorics},
     pages = {261--287},
     year = {2026},
     publisher = {The Combinatorics Consortium},
     volume = {9},
     number = {1},
     doi = {10.5802/alco.469},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.469/}
}
TY  - JOUR
AU  - Pantangi, Venkata Raghu Tej
AU  - Sin, Peter
TI  - Perfect state transfer in graphs related to linear groups in two dimensions
JO  - Algebraic Combinatorics
PY  - 2026
SP  - 261
EP  - 287
VL  - 9
IS  - 1
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.469/
DO  - 10.5802/alco.469
LA  - en
ID  - ALCO_2026__9_1_261_0
ER  - 
%0 Journal Article
%A Pantangi, Venkata Raghu Tej
%A Sin, Peter
%T Perfect state transfer in graphs related to linear groups in two dimensions
%J Algebraic Combinatorics
%D 2026
%P 261-287
%V 9
%N 1
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.469/
%R 10.5802/alco.469
%G en
%F ALCO_2026__9_1_261_0
Pantangi, Venkata Raghu Tej; Sin, Peter. Perfect state transfer in graphs related to linear groups in two dimensions. Algebraic Combinatorics, Volume 9 (2026) no. 1, pp. 261-287. doi: 10.5802/alco.469

[1] Ahmadi, Bahman; Shirdareh Haghighi, M. H.; Mokhtar, Ahmad Perfect quantum state transfer on the Johnson scheme, Linear Algebra Appl., Volume 584 (2020), pp. 326-342 | DOI | MR | Zbl

[2] Angeles-Canul, Ricardo J.; Norton, Rachael M.; Opperman, Michael C.; Paribello, Christopher C.; Russell, Matthew C.; Tamon, Christino Perfect state transfer, integral circulants, and join of graphs, Quantum Inf. Comput., Volume 10 (2010) no. 3-4, pp. 325-342 | MR | Zbl

[3] Árnadóttir, Arnbjörg Soffía; Godsil, Chris On state transfer in Cayley graphs for abelian groups, Quantum Inf. Process., Volume 22 (2023) no. 1, Paper no. 8, 24 pages | DOI | MR | Zbl

[4] Bannai, Eiichi; Ito, Tatsuro Algebraic combinatorics. I. Association schemes, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984, xxiv+425 pages | MR | Zbl

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

[6] Bernasconi, Anna; Godsil, Chris; Severini, Simone Quantum networks on cubelike graphs, Phys. Rev. A (3), Volume 78 (2008) no. 5, Paper no. 052320, 5 pages | DOI | MR

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

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

[9] Cheung, Wang-Chi; Godsil, Chris Perfect state transfer in cubelike graphs, Linear Algebra Appl., Volume 435 (2011) no. 10, pp. 2468-2474 | DOI | MR | Zbl

[10] Christandl, Matthias; Datta, Nilanjana; Ekert, Artur; Landahl, Andrew J. Perfect State Transfer in Quantum Spin Networks, Phys. Rev. Lett., Volume 92 (2004), p. 187902 | DOI

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

[12] Coutinho, Gabriel; Godsil, Chris Perfect state transfer in products and covers of graphs, Linear Multilinear Algebra, Volume 64 (2016) no. 2, pp. 235-246 | DOI | MR | Zbl

[13] Coutinho, Gabriel; Godsil, Chris Graph Spectra and Continuous Quantum Walks (2021) https://www.math.uwaterloo.ca/... (Accessed 2025-11-14 unpublished)

[14] 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

[15] Coutinho, Gabriel; Juliano, Emanuel; Jung Spier, Thomás No perfect state transfer in trees with more than 3 vertices, J. Combin. Theory Ser. B, Volume 168 (2024), pp. 68-85 | DOI | MR | Zbl

[16] Ennola, Veikko On the conjugacy classes of the finite unitary groups, Ann. Acad. Sci. Fenn. Ser. A I, Volume 313 (1962), p. 13 | MR | Zbl

[17] Ennola, Veikko On the characters of the finite unitary groups, Ann. Acad. Sci. Fenn. Ser. A I, Volume 323 (1963), p. 35 | MR | Zbl

[18] Fulton, William; Harris, Joe Representation theory. A First Course., Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991, xvi+551 pages | DOI | MR | Zbl

[19] Godsil, Chris State transfer on graphs, Discrete Math., Volume 312 (2012) no. 1, pp. 129-147 | DOI | MR | Zbl

[20] Godsil, Chris; Meagher, Karen Erdős–Ko–Rado Theorems: Algebraic Approaches, Cambridge Studies in Advanced Mathematics, 149, Cambridge University Press, Cambridge, 2016, xvi+335 pages | DOI | MR | Zbl

[21] Gow, Roderick Two multiplicity-free permutation representations of the general linear group GL (n,q 2 ), Math. Z., Volume 188 (1984) no. 1, pp. 45-54 | DOI | MR | Zbl

[22] Jordan, Herbert E. Group-Characters of Various Types of Linear Groups, Amer. J. Math., Volume 29 (1907) no. 4, pp. 387-405 | DOI | MR | Zbl

[23] Kay, Alastair Basics of perfect communication through quantum networks, Phys. Rev. A, Volume 84 (2011), Paper no. 022337, 14 pages | DOI | Zbl

[24] Schur, J. Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math., Volume 132 (1907), pp. 85-137 | DOI | MR | Zbl

[25] Sin, Peter; Sorci, Julien Continuous-time quantum walks on Cayley graphs of extraspecial groups, Algebr. Comb., Volume 5 (2022) no. 4, pp. 699-714 | DOI | MR | Zbl | Numdam

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

[27] Vinet, Luc; Zhan, Hanmeng Perfect state transfer on weighted graphs of the Johnson scheme, Letters in Mathematical Physics, Volume 110 (2020) no. 9, pp. 2491-2504 | DOI | MR | Zbl

Cited by Sources: