Quantum walks on embeddings

Abstract

We introduce a new type of discrete quantum walks, called vertex-face walks, based on orientable embeddings. We first establish a spectral correspondence between the transition matrix U and the vertex-face incidence structure. Using the incidence graph, we derive a formula for the principal logarithm of \(U^2\), and find conditions for its underlying digraph to be an oriented graph. In particular, we show this happens if the vertex-face incidence structure forms a partial geometric design. We also explore properties of vertex-face walks on the covers of a graph. Finally, we study a non-classical behavior of vertex-face walks.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

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)

  2. 2.

    Ambainis, A., Portugal, R., Nahimov, N.: Spatial search on grids with minimum memory. Quant. Inf. Comput. 15(13–14), 1233–1247 (2015)

    MathSciNet  Google Scholar 

  3. 3.

    Biggs, N.: Automorphisms of imbedded graphs. J. Comb. Theory Ser. B 11(2), 132–138 (1971)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bose, R.C., Shrikhande, S.S., Singhi, N.M.: Edge regular multigraphs and partial geometric designs with an application to the embedding of quasi-regular designs. Int. Colloq. Comb. Theory 1, 49 (1973)

    MATH  Google Scholar 

  5. 5.

    Bridges, W.G., Shrikhande, M.S.: Special partially balanced incomplete block designs and associated graphs. Discrete Math. 9(1), 1–18 (1974)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Childs, A.: Universal computation by quantum walk. Phys. Rev. Lett. 102(18), 180501 (2009)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Falk, M.: Quantum Search on the Spatial Grid, arXiv:1303.4127 (2013)

  8. 8.

    Godsil, C.: Algebraic Combinatorics. Chapman & Hall, London (1993)

    Google Scholar 

  9. 9.

    Godsil, C.: Periodic graphs. Electron. J. Combin. 18(1), P23 (2011)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Godsil, C.: Graph Spectra and Quantum Walks. PIMS, Vancover (2015)

    Google Scholar 

  11. 11.

    Godsil, C.: Real State Transfer, arXiv:1710.04042 (2017)

  12. 12.

    Godsil, C.: Sedentary Quantum Walks, arXiv:1710.11192 (2017)

  13. 13.

    Gross, J., Tucker, T.: Topological Graph Theory. Dover Publications, Mineola (2001)

    Google Scholar 

  14. 14.

    Gustin, W.: Orientable embedding of Cayley graphs. Bull. Amer. Math. Soc. 69(2), 272–276 (1963)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  16. 16.

    Heffter, L.: Ueber Tripelsysteme. Mathematische Annalen 49(1), 101–112 (1897)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kendon, V.: Quantum walks on general graphs. Int. J. Quant. Inf. 4(05), 791–805 (2006)

    Article  Google Scholar 

  18. 18.

    Kendon, V., Tamon, C.: Perfect state transfer in quantum walks on graphs. Quant. Inf. Comput. 14, 417–438 (2014)

    Google Scholar 

  19. 19.

    Konno, N., Portugal, R., Sato, I., Segawa, E.: Partition-based discrete-time quantum walks. Quant. Inf. Process. 17(4), 100 (2018)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Lato, S.: Quantum Walks on Oriented Graphs, Ph.D. thesis, University of Waterloo (2019)

  21. 21.

    Lins, S.: Graph-encoded maps. J. Combin. Theory Ser. B 32(2), 171–181 (1982)

    MathSciNet  Article  Google Scholar 

  22. 22.

    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)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Patel, A., Raghunathan, K., Rungta, P.: Quantum random walks do not need a coin toss. Phys. Rev. A 71, 032347 (2005)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Petroelje, W.S.: ScholarWorks at WMU Imbedding Graphs in Pseudosurfaces, Ph.D. thesis, Western Michigan University (1971)

  25. 25.

    Portugal, R., Santos, R., Fernandes, T., Gonçalves, D.: The staggered quantum walk model. Quant. Inf. Process. 15(1), 85–101 (2016)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Ringel, G.: Über das Problem der Nachbargebiete auf orientierbaren Flächen. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 25(1–2), 105–127 (1961)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 32–41 (2004)

  28. 28.

    Terry, C.M., Welch, L.R., Youngs, J.W.T.: The genus of K12s. J. Combin. Theory 2(1), 43–60 (1967)

    Article  Google Scholar 

  29. 29.

    Underwood, M., Feder, D.: Universal quantum computation by discontinuous quantum walk. Phys. Rev. A 82(4), 042304 (2010)

    MathSciNet  Article  Google Scholar 

  30. 30.

    van Dam, E.R., Spence, E.: Combinatorial designs with two singular values–I: uniform multiplicative designs. J. Combin. Theory Ser. A 107(1), 127–142 (2004)

    MathSciNet  Article  Google Scholar 

  31. 31.

    van Dam, E.R., Spence, E.: Combinatorial designs with two singular values II. Partial geometric designs. Linear Algebra Appl. 396, 303–316 (2005)

    MathSciNet  Article  Google Scholar 

  32. 32.

    White, A.: Graphs, Groups, and Surfaces. North-Holland, Amsterdam (1984)

    Google Scholar 

  33. 33.

    Wong, T.G.: Equivalence of Szegedy’s and coined quantum walks. Quantum Inf. Process. 16(9), 215 (2017). https://doi.org/10.1007/s11128-017-1667-y

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Zhan, H.: Discrete Quantum Walks on Graphs and Digraphs, Ph.D. thesis (2018)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hanmeng Zhan.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhan, H. Quantum walks on embeddings. J Algebr Comb (2020). https://doi.org/10.1007/s10801-020-00958-z

Download citation

Keywords

  • Quantum walks
  • Graph embeddings
  • Incidence matrices
  • Graph spectra

Mathematics Subject Classification

  • 05C50
  • 05E99