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An infinite family of circulant graphs with perfect state transfer in discrete quantum walks

  • Hanmeng ZhanEmail author
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Abstract

We study perfect state transfer in Kendon’s model of discrete quantum walks. In particular, we give a characterization of perfect state transfer purely in terms of the graph spectra, and construct an infinite family of 4-regular circulant graphs that admit perfect state transfer. Prior to our work, the only known infinite families of examples were variants of cycles and diamond chains.

Notes

Acknowledgements

The author would like to thank Ada Chan, Gabriel Coutinho, Chris Godsil, Krystal Guo, and Christino Tamon for their helpful discussion and generous comments. The author appreciates the time and effort taken by the referees to review this manuscript.

References

  1. 1.
    Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 50–59 (2001)Google Scholar
  2. 2.
    Angeles-Canul, R., Norton, R., Opperman, M., Paribello, C., Russell, M., Tamon, C.: On quantum perfect state transfer in weighted join graphs. Int. J. Quantum Inf. 7(8), 1429–1445 (2009)zbMATHGoogle Scholar
  3. 3.
    Angeles-Canul, R., Norton, R., Opperman, M., Paribello, C., Russell, M., Tamon, C.: Perfect state transfer, integral circulants and join of graphs. Quantum Inf. Comput. 10, 325–342 (2010) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bachman, R., Fredette, E., Fuller, J., Landry, M., Opperman, M., Tamon, C., Tollefson, A.: Perfect state transfer on quotient graphs. Quantum Inf. Comput. 12, 293–313 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Banchi, L., Coutinho, G., Godsil, C., Severini, S.: Pretty good state transfer in qubit chains–the Heisenberg Hamiltonian. J. Math. Phys. 58(3), 032202 (2017) ADSMathSciNetzbMATHGoogle Scholar
  6. 6.
    Barr, K., Proctor, T., Allen, D., Kendon, V.: Periodicity and perfect state transfer in quantum walks on variants of cycles. Quantum Inf. Comput. 14, 417–438 (2014)MathSciNetGoogle Scholar
  7. 7.
    Bose, S.: Quantum communication through an unmodulated spin chain. Phys. Rev. Lett. 91, 207901 (2003) ADSGoogle Scholar
  8. 8.
    Cheung, W., Godsil, C.: Perfect state transfer in cubelike graphs. Linear Algebra Appl. 435, 2468–2474 (2011)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Childs, A.: Universal computation by quantum walk. Phys. Rev. Lett. 102(18), 180501 (2009)ADSMathSciNetGoogle Scholar
  10. 10.
    Christandl, M., Datta, N., Ekert, A., Landahl, A.J.: Perfect state transfer in quantum spin networks. Phys. Rev. Lett. 92(18), 187902 (2004)ADSGoogle Scholar
  11. 11.
    Christandl, M., Datta, N., Ekert, A., Landahl, A.J.: Perfect state transfer in quantum spin networks. Phys. Rev. Lett. 92(18), 187902 (2004)ADSGoogle Scholar
  12. 12.
    Coutinho, G.: Quantum state transfer in graphs, Ph.D. thesis, University of Waterloo (2014)Google Scholar
  13. 13.
    Coutinho, G., Godsil, C.: Perfect state transfer in products and covers of graphs. Linear Multilinear Algebra 64, 1–12 (2015)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Coutinho, G., Godsil, C., Guo, K., Vanhove, F.: Perfect state transfer on distance-regular graphs and association schemes. Linear Algebra Appl. 478, 108–130 (2015)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Coutinho, G., Guo, K., Van Bommel, C.M.: Pretty good state transfer between internal nodes of paths. Quantum Inf. Comput. 17(9–10), 825–830 (2017)MathSciNetGoogle Scholar
  16. 16.
    Coutinho, G., Liu, H.: No Laplacian perfect state transfer in trees. SIAM J. Discret. Math. 29(4), 2179–2188 (2015)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Eisenberg, O., Kempton, M., Lippner, G.: Pretty good quantum state transfer in asymmetric graphs via potential. Discret. Math. 342, 2821 (2018).  https://doi.org/10.1016/j.disc.2018.10.037 MathSciNetzbMATHGoogle Scholar
  18. 18.
    Godsil, C.: Periodic graphs. Electron. J. Comb. 18(1), 23 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Godsil, C.: Graph Spectra and Quantum Walks, Unpublished, (2015)Google Scholar
  20. 20.
    Godsil, C., Royle, G.: Algebraic Graph Theory. Springer, New York (2001)zbMATHGoogle Scholar
  21. 21.
    Godsil, C., Smith, J.: Strongly cospectral vertices, arXiv:1709.07975 (2017)
  22. 22.
    Kay, A.: Perfect state transfer: beyond nearest-neighbor couplings. Phys. Rev. A 73(3), 032306 (2006)ADSMathSciNetGoogle Scholar
  23. 23.
    Kay, A.: Basics of perfect communication through quantum networks. Phys. Rev. A 84(2), 022337 (2011)ADSGoogle Scholar
  24. 24.
    Kempton, M., Lippner, G., Yau, S.T.: Pretty good quantum state transfer in symmetric spin networks via magnetic field. Quantum Inf. Process. 16(9), 1–23 (2017)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Kendon, V.: Quantum walks on general graphs. Int. J. Quantum Inf. 4, 791–805 (2006)zbMATHGoogle Scholar
  26. 26.
    Kendon, V., Tamon, C.: Perfect state transfer in quantum walks on graphs. Quantum Inf. Comput. 14, 417–438 (2014)MathSciNetGoogle Scholar
  27. 27.
    Kurzyński, P., Wójcik, A.: Discrete-time quantum walk approach to state transfer. Phys. Rev. A Atomic Mol. Opt. Phys. 83(6), 062315 (2011)Google Scholar
  28. 28.
    Lovett, N., Cooper, S., Everitt, M., Trevers, M., Kendon, V.: Universal quantum computation using the discrete-time quantum walk. Phys. Rev. A 81(4), 042330 (2010)ADSMathSciNetGoogle Scholar
  29. 29.
    Portugal, R., Santos, R., Fernandes, T., Gonçalves, D.: The staggered quantum walk model. Quantum Inf. Process. 15(1), 85–101 (2016)ADSMathSciNetzbMATHGoogle Scholar
  30. 30.
    Štefaňák, M., Skoupý, S.: Perfect state transfer by means of discrete-time quantum walk search algorithms on highly symmetric graphs. Phys. Rev. A 94(2), 022301 (2016)ADSzbMATHGoogle Scholar
  31. 31.
    Štefaňák, M., Skoupý, S.: Perfect state transfer by means of discrete-time quantum walk on complete bipartite graphs. Quantum Inf. Process. 16(3), 72 (2017)ADSMathSciNetzbMATHGoogle Scholar
  32. 32.
    Szegedy, M.: Quantum speed-up of Markov chain based algorithms. 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 32–41 (2004)Google Scholar
  33. 33.
    Underwood, M., Feder, D.: Universal quantum computation by discontinuous quantum walk. Phys. Rev. A 82(4), 1–69 (2010)MathSciNetGoogle Scholar
  34. 34.
    Yalçınkaya, I., Gedik, Z.: Qubit state transfer via discrete-time quantum walks. J. Phys. A Math. Theor. 48(22), 225302 (2015)ADSMathSciNetzbMATHGoogle Scholar
  35. 35.
    Yoshie, Y.: Periodicity of Grover walks on distance-regular graphs. arXiv:1805.07681 (2018)
  36. 36.
    Zhan, H.: Discrete Quantum Walks on Graphs and Digraphs, Ph.D. thesis (2018)Google Scholar
  37. 37.
    Zhan, X., Qin, H., Bian, Z., Li, J., Xue, P.: Perfect state transfer and efficient quantum routing: a discrete-time quantum walk approach. Phys. Rev. A 90, 012331 (2014)ADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  2. 2.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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