Fractional revival on integral mixed circulant graphs
Abstract
Fractional revival on graphs can be utilized to transfer quantum information between set of distinct nodes in a quantum spin network. In this work, we prove existence of fractional revival on integral mixed circulant graphs. Spectral characterization and number-theoretic condition for integral mixed circulant graphs is proposed, to possess fractional revival between two distinct nodes, we also present examples in support of the theorem.
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References
Abdollahi, A., Vatandoost, E., Which Cayley graphs are integral?., The Electronic Journal of Combinatorics, 16(1),
(2009).
Bernard, P. A., Chan, A., Loranger, E., Tamon, C., Vinet, L., A graph with fractional revival, Physics Letters A, 382(5), 259-264, (2018).
Chan, A., Coutinho, G., Tamon, C., Vinet, L., Zhan, H., Quantum fractional revival on graphs, Discrete Applied Mathematics, 269, 86–98, (2019).
Chan, A., Coutinho, G., Tamon, C., Vinet, L., Zhan, H. , Fractional revival and association schemes, Discrete Mathematics, 343 (11), 112018, (2020).
Chan, A., Coutinho, G., Drazen, W., Eisenberg, O., Godsil, C., Kempton, M., Lippner, G., Tamon, C. Zhan, H., Fundamentals of fractional revival in graphs., Linear Algebra and its Applications, 655, 129-158, (2022).
Monakhova, E. A., Romanov, A. Y., Lezhnev, E. V., Shortest path search algorithm in optimal two-dimensional circulant networks: Implementation for networks-on-chip, IEEE Access, 8, 215010-215019, (2020).
Elspas, B., Turner, J.. Graphs with circulant adjacency matrices, Journal of Combinatorial Theory, 9(3), 297-307, (1970).
Steinberg, B., Representation theory of finite groups: an introductory approach, New York: Springer, (2012).
Godsil, C., State transfer on graphs, Discrete Math., 312, 123–147, (2012).
Meyer, C. D., Matrix analysis and applied linear algebra, (Vol. 188). Siam, (2023).
Coutinho, G., Quantum state transfer in graphs, PhD thesis, University of Waterloo, 2014.
Coutinho, G., Liu, H., No Laplacian perfect state transfer in trees, SIAM Journal on Discrete Mathematics, 29(4), 2179-2188, (2015).
Thongsomnuk, I., Meemark, Y., Perfect state transfer in unitary Cayley graphs and gcd-graphs, Linear and Multilinear Algebra, 67(1), 39-50, (2019).
Bang-Jensen, J., Gutin, G. Z., Digraphs: Theory, Algorithms and Applications, Springer Science Business Media, (2008).
Liu, J., Li, X., Hermitian-adjacency matrices and Hermitian energies of mixed graphs, Linear Algebra and its Applications, 466, 182-207, (2015).
Wang, J., Wang, L., Liu, X., Fractional revival on Cayley graphs over abelian groups, arXiv preprint arXiv:2206.12584, (2022).
Zhou, J., Bu, C., Shen, J., Some results for the periodicity and perfect state transfer, The Electronic Journal of Combinatorics, P184-P184, (2011).
Babai, L., Spectra of Cayley graphs. Journal of Combinatorial Theory, Series B, 27(2), 180-189, (1979).
Basic, M., Petkovic, M. D., Stevanovi´c, D., Perfect state transfer in integral circulant graphs, Applied Mathematics Letters, 22(7), 1117-1121, (2009).
Basic, M., Characterization of quantum circulant networks having perfect state transfer, Quantum Information Processing, 12(1), 345-364, (2013).
Christandl, M., Datta, N., Dorlas, T. C., Ekert, A., Kay, A., Landahl, A. J.,Perfect transfer of arbitrary states in quantum spin networks, Physical Review A, 71(3), 032312, (2005).
Christandl, M., Datta, N., Ekert, A., Landahl, A. J., Perfect state transfer in quantum spin networks, Physical review letters, 92(18), 187902, (2004).
Kadyan, M., Bhattacharjya, B., Integral mixed circulant graphs, Discrete Math., 346 (1), 113142, (2023).
Biggs, N., Algebraic Graph Theory (No. 67). Cambridge university press, (1993).
Cayley, P., Desiderata and suggestions: No. 2. The Theory of groups: graphical representation, American journal of mathematics, 1(2), 174-176, (1878).
Angeles-Canul, R. J., Norton, R. M., Opperman, M. C., Paribello, C. C., Russell, M. C., Tamon, C., Perfect state transfer, integral circulants, and join of graphs, Quantum Information and Computation, 10(3-4), 325-342, (2010).
Bose, S., Quantum Communication through an Unmodulated Spin Chain, Phys. Rev. Lett., 91 (20), 207901, (2003).
So, W., Integral circulant graphs, Discrete Math, 306 (1), 153-158, (2006).
Genest, V. X., Vinet, L., Zhedanov, A., Quantum spin chains with fractional revival, Annals of Physics, 371, 348-367, (2016).
Cao, X., Luo, G., Fractional revival on abelian Cayley graphs, arXiv preprint arXiv:2208.05107, (2022).
Song, X., Quantum state transfer on integral oriented circulant graphs, Applied Mathematics and Computation, 464, 128391, (2024).
Song, X., Lin, H., State transfer on integral mixed circulant graphs, Discrete Mathematics, 347(1), 113727, (2024).
Liu, X., Zhou, S., Eigenvalues of Cayley Graphs, The Electronic Journal of Combinatorics, P2-9, (2022).
Tan, Y. Y., Feng, K., Cao, X., Perfect state transfer on abelian Cayley graphs, Linear Algebra and its Applications, 563, 331-352, (2019).
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