Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Quantum state transfer between twins in weighted graphs

  • Published:
Journal of Algebraic Combinatorics Aims and scope Submit manuscript

Abstract

Twin vertices in simple unweighted graphs are vertices that have the same neighbours, and, in the case of weighted graphs with possible loops, the corresponding incident edges have equal weights. In this paper, we explore the role of twin vertices in quantum state transfer. In particular, we provide characterizations of periodicity, perfect state transfer, and pretty good state transfer between twin vertices in a weighted graph with respect to its adjacency, Laplacian and signless Laplacian matrices. As an application, we provide characterizations of all simple unweighted double cones on regular graphs that exhibit periodicity, perfect state transfer, and pretty good state transfer.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Data availability

The study has no associated data.

References

  1. Alvir, R., Dever, S., Lovitz, B., Myer, J., Tamon, C., Xu, Y., Zhan, H.: Perfect state transfer in Laplacian quantum walk. J. Algebraic Combin. 43(4), 801–826 (2016). https://doi.org/10.1007/s10801-015-0642-x

    Article  MathSciNet  MATH  Google Scholar 

  2. Angeles-Canul, R., Norton, R., Opperman, M., Paribello, C., Russell, M., Tamon, C.: Quantum perfect state transfer on weighted join graphs. Int. J. Quantum Inf. 07(08), 1429–1445 (2009)

    Article  MATH  Google Scholar 

  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(3–4), 0325–0342 (2010). https://doi.org/10.26421/qic10.3-4-10

    Article  MathSciNet  MATH  Google Scholar 

  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(3–4), 293–313 (2012). https://doi.org/10.26421/qic12.3-4-9

    Article  MathSciNet  MATH  Google Scholar 

  5. Banchi, L., Coutinho, G., Godsil, C., Severini, S.: Pretty good state transfer in qubit chains—The Heisenberg Hamiltonian. J. Math. Phys. 58(3), 32202 (2017). https://doi.org/10.1063/1.4978327

    Article  MathSciNet  MATH  Google Scholar 

  6. Bašić, M., Petković, M., Stevanović, D.: Perfect state transfer in integral circulant graphs. Appl. Math. Lett. 22(7), 1117–1121 (2009). https://doi.org/10.1016/j.aml.2008.11.005

    Article  MathSciNet  MATH  Google Scholar 

  7. Bose, S.: Quantum communication through an unmodulated spin chain. Phys. Rev. Lett. 91(20), 207901 (2003)

    Article  Google Scholar 

  8. Bose, S., Casaccino, A., Mancini, S., Severini, S.: Communication in XYZ all-to-all quantum networks with a missing link. Int. J. Quantum Inf. 7(4), 713–723 (2009). https://doi.org/10.1142/S0219749909005389

    Article  MATH  Google Scholar 

  9. Cao, X., Wang, D., Feng, K.: Pretty good state transfer on Cayley graphs over dihedral groups. Discrete Math. 343(1), 111636 (2020). https://doi.org/10.1016/j.disc.2019.111636

    Article  MathSciNet  MATH  Google Scholar 

  10. Casaccino, A., Lloyd, S., Mancini, S., Severini, S.: Quantum state transfer through a qubit network with energy shifts and fluctuations. Int. J. Quantum Inf. 7(8), 1417–1427 (2009). https://doi.org/10.1142/S0219749909006085

    Article  MATH  Google Scholar 

  11. Chen, Q., Godsil, C.D.: Pair state transfer. Quantum Inf. Process. 19, 321 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cheung, W.C., Godsil, C.: Perfect state transfer in cubelike graphs. Linear Algebra Appl. 435(10), 2468–2474 (2011). https://doi.org/10.1016/j.laa.2011.04.022

    Article  MathSciNet  MATH  Google Scholar 

  13. Christandl, M., Datta, N., Ekert, A., Landahl, A.J.: Perfect state transfer in quantum spin networks. Phys. Rev. Lett. 92(18), 187902 (2004)

    Article  Google Scholar 

  14. Christandl, M., Datta, N., Dorlas, T.C., Ekert, A., Kay, A., Landahl, A.J.: Perfect transfer of arbitrary states in quantum spin networks. Phys. Rev. A 71(3), 032312 (2005). https://doi.org/10.1103/PhysRevA.71.032312

    Article  Google Scholar 

  15. Coutinho, G., Godsil, C.: Graph spectra and continuous quantum walks. https://www.math.uwaterloo.ca/~cgodsil/quagmire/pdfs/GrfSpc3.pdf (2021)

  16. Coutinho, G.: Quantum State Transfer in Graphs. PhD. Dissertation (2014)

  17. Coutinho, G., Liu, H.: No Laplacian perfect state transfer in trees. SIAM J. Discrete Math. 29(4), 2179–2188 (2015). https://doi.org/10.1137/140989510

    Article  MathSciNet  MATH  Google Scholar 

  18. 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). https://doi.org/10.1016/j.laa.2015.03.024

    Article  MathSciNet  MATH  Google Scholar 

  19. Eisenberg, O., Kempton, M., Lippner, G.: Pretty good quantum state transfer in asymmetric graphs via potential. Discrete Math. 342(10), 2821–2833 (2019). https://doi.org/10.1016/j.disc.2018.10.037

    Article  MathSciNet  MATH  Google Scholar 

  20. Fan, X., Godsil, C.: Pretty good state transfer on double stars. Linear Algebra Appl. 438(5), 2346–2358 (2013). https://doi.org/10.1016/j.laa.2012.10.006

    Article  MathSciNet  MATH  Google Scholar 

  21. Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58(2), 915–928 (1998). https://doi.org/10.1103/PhysRevA.58.915. ([quant-ph])

    Article  MathSciNet  Google Scholar 

  22. Ge, Y., Greenberg, B., Perez, O., Tamon, C.: Perfect state transfer, graph products and equitable partitions. Int. J. Quantum Inf. 9(3), 823–842 (2011). https://doi.org/10.1142/S0219749911007472

    Article  MathSciNet  MATH  Google Scholar 

  23. Godsil, C., Smith, J.: Strongly cospectral vertices. arXiv:1709.07975 [math.CO] (2017)

  24. Godsil, C.: Periodic graphs. Electron. J. Combin. 18(1), 1–15 (2011). https://doi.org/10.37236/510

    Article  MathSciNet  MATH  Google Scholar 

  25. Godsil, C.: State transfer on graphs. Discrete Math. 312(1), 129–147 (2012). https://doi.org/10.1016/j.disc.2011.06.032

    Article  MathSciNet  MATH  Google Scholar 

  26. Godsil, C.: When can perfect state transfer occur? Electron. J. Linear Algebra 23, 877–890 (2012). https://doi.org/10.13001/1081-3810.1563

    Article  MathSciNet  MATH  Google Scholar 

  27. Godsil, C., Royle, G.: Algebraic graph theory. Graduate texts in mathematics, vol. 207. Springer (2001)

    Book  MATH  Google Scholar 

  28. Godsil, C., Kirkland, S., Severini, S., Smith, J.: Number-theoretic nature of communication in quantum spin systems. Phys. Rev. Lett. 109(5), 3–6 (2012). https://doi.org/10.1103/PhysRevLett.109.050502

    Article  Google Scholar 

  29. Horn, R., Johnson, C.: Topics in matrix analysis. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  30. Horn, R., Johnson, C.: Matrix analysis, 2nd edn. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  31. Johnston, N., Kirkland, S., Plosker, S., Storey, R., Zhang, X.: Perfect quantum state transfer using Hadamard diagonalizable graphs. Linear Algebra Appl. 531, 375–398 (2017). https://doi.org/10.1016/j.laa.2017.05.037

    Article  MathSciNet  MATH  Google Scholar 

  32. Kay, A.: Perfect, efficient, state transfer and its application as a constructive tool. Int. J. Quantum Inf. 8(4), 641–676 (2010). https://doi.org/10.1142/S0219749910006514

    Article  MATH  Google Scholar 

  33. Kay, A.: Basics of perfect communication through quantum networks. Phys. Rev. A 84(2), 022337 (2011). https://doi.org/10.1103/PhysRevA.84.022337

    Article  Google Scholar 

  34. Kempton, M., Lippner, G., Yau, S.-T.: Pretty good state transfer in symmetric spin networks via magnetic field. Quantum Inf. Process. 16(9), 1–23 (2017). https://doi.org/10.1007/s11128-017-1658-z

  35. Kempton, M., Lippner, G., Yau, S.-T.: Perfect state transfer on graphs with a potential. Quantum Inf. Comput. 17(3–4), 0303–0327 (2017). https://doi.org/10.26421/qic17.3-4-7. arXiv:1611.02093

    Article  MathSciNet  Google Scholar 

  36. Kendon, V.: Quantum walks on general graphs. Int. J. Quantum Inf. 4(5), 791–805 (2006). https://doi.org/10.1142/S0219749906002195. ([quant-ph])

    Article  MATH  Google Scholar 

  37. Kendon, V., Tamon, C.: Perfect state transfer in quantum walks on graphs. J. Comput. Theor. Nanosci. 8(3), 422–433 (2011). https://doi.org/10.1166/jctn.2011.1706

    Article  Google Scholar 

  38. Kirkland, S., Severini, S.: Spin-system dynamics and fault detection in threshold networks. Phys. Rev. A 83, 012310 (2011). https://doi.org/10.1103/PhysRevA.83.012310

    Article  Google Scholar 

  39. Li, Y., Liu, X., Zhang, S., Zhou, S.: Perfect state transfer in NEPS of complete graphs. Discrete Appl. Math. 289, 98–114 (2021). https://doi.org/10.1016/j.dam.2020.09.024

    Article  MathSciNet  MATH  Google Scholar 

  40. Monterde, H.: Quantum state transfer between twins in graphs. MSc. Thesis (2021) http://hdl.handle.net/1993/35937

  41. Monterde, H.: Strong cospectrality and twin vertices in weighted graphs. Electron. J. Linear Algebra 38, 494–518 (2021)

    MathSciNet  MATH  Google Scholar 

  42. Pal, H.: Quantum state transfer on a class of circulant graphs. Linear Multilinear Algebra 69, 2527–2538 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  43. Pal, H.: Laplacian state transfer on graphs with an edge perturbation between twin vertices. Discrete Math. 345(7), 112872 (2022). https://doi.org/10.1016/j.disc.2022.112872

    Article  MathSciNet  MATH  Google Scholar 

  44. Pal, H., Bhattacharjya, B.: Pretty good state transfer on some NEPS. Discrete Math. 340(4), 746–752 (2017). https://doi.org/10.1016/j.disc.2016.11.026

    Article  MathSciNet  MATH  Google Scholar 

  45. Pal, H., Bhattacharjya, B.: Pretty good state transfer on circulant graphs. Electron. J. Combin. (2017). https://doi.org/10.37236/6388

    Article  MathSciNet  MATH  Google Scholar 

  46. van Bommel, C.: A complete characterization of pretty good state transfer on paths. Quantum Inf. Comput. 19(7–8), 601–608 (2019). https://doi.org/10.26421/qic19.7-8-5

    Article  MathSciNet  Google Scholar 

  47. Vinet, L., Zhedanov, A.: Almost perfect state transfer in quantum spin chains. Phys. Rev. A 86(5), 052319 (2012). https://doi.org/10.1103/PhysRevA.86.052319

    Article  Google Scholar 

Download references

Acknowledgements

H.M. is supported by the University of Manitoba Faculty of Science and Faculty of Graduate Studies. S.K. is supported by NSERC Discovery Grant RGPIN-2019-05408. S.P. is supported by NSERC Discovery Grant Number 1174582, the Canada Foundation for Innovation (CFI) Grant Number 35711, and the Canada Research Chairs (CRC) Program Grant Number 231250. We thank the referees for their comments and suggestions that helped improved this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada

    Steve Kirkland, Hermie Monterde & Sarah Plosker

  2. Department of Mathematics and Computer Science, University of Manitoba, Brandon, MB, R7A 6A9, Canada

    Sarah Plosker

Authors
  1. Steve Kirkland
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hermie Monterde
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sarah Plosker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hermie Monterde.

Ethics declarations

Conflict of interest

The authors declare that no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kirkland, S., Monterde, H. & Plosker, S. Quantum state transfer between twins in weighted graphs. J Algebr Comb 58, 623–649 (2023). https://doi.org/10.1007/s10801-023-01261-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10801-023-01261-3

Keywords

Mathematics Subject Classification

Search

Navigation