Skip to main content
SpringerLink
Log in

Quantum Walks on Two Kinds of Two-Dimensional Models

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

In this paper, we numerically study quantum walks on two kinds of two-dimensional graphs: cylindrical strip and Mobius strip. The two kinds of graphs are typical two-dimensional topological graph. We study the crossing property of quantum walks on these two models. Also, we study its dependence on the initial state, size of the model. At the same time, we compare the quantum walk and classical walk on these two models to discuss the difference of quantum walk and classical walk.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Wu, D.C., Liao, Q.: Random walk terminally attached to wall on different lattices. Chin. Sci. Bull. 42(5), 433–437 (1997)

    Article  MathSciNet  Google Scholar 

  2. Cheng, X.l., Hu, F., Zeng, Q.C.: Stochastic method to determine the scale and anomalous diffusion of gusts in a windy atmospheric boundary layer. Chin. Sci. Bull. 59(34), 4890–4898 (2014)

    Article  Google Scholar 

  3. An, T.L., Xiao, L.Z., Li, X., etc.: Investigation of the correlation between internal gradients and dephasing effect in inhomogeneous field. Sci. China Phys. Mech. Astron. 57(9), 1676–1683 (2014)

    Article  ADS  Google Scholar 

  4. Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quant. Inf. Proc. 11(5), 1015–1106

  5. Wang, C., Li, Y.S., Hao, L.: Optical implementation of quantum random walks using weak cross-Kerr media. Chin. Sci. Bull. 56(20), 2088–2091 (2011)

    Article  Google Scholar 

  6. Hamilton, C.S., Gabris, A., Jex, I., Barnett, S.M.: Quantum walk with a four-dimensional coin. New J. Phys. 13, 013015 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  7. Ambainis, A.: Quantum walk algorithm for element distinctness. SIAM J. Comput. 37(1), 210–239 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Shenvi, N., Kempe, J., Whaley, K.B.: Quantum random-walk search algorithm. Phys. Rev. A 67, 052307 (2003)

    Article  ADS  Google Scholar 

  9. Hein, B., Tanner, G.: Quantum search algorithms on a regular lattice. Phys. Rev. A 82, 012326 (2010)

    Article  ADS  Google Scholar 

  10. Berry, S.D., Wang, J.B.: Quantum-walk-based search and centrality. Phys. Rev. A 82, 042333 (2010)

    Article  ADS  Google Scholar 

  11. Tarrataca, L., Wichert, A.: Intricacies of quantum computational paths. Quant. Inf. Proc. 12(2), 1365–1378 (2013)

    Article  MATH  Google Scholar 

  12. Li, D., Zhang, J., Guo, F.Z., Huang, W., Wen, Q.Y., Chen, H.: Discrete-time interacting quantum walks and quantum Hash schemes. Quant. Inf. Proc. 12(3), 1501–1513 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, D., Zhang, J., Ma, X.W., Zhang, W.W., Wen, Q.Y.: Analysis of the two-particle controlled interacting quantum walks. Quant. Inf. Proc. 12(6), 2167–2176 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Berry, S.D., Wang, J.B.: Two-particle quantum walks: Entanglement and graph isomorphism testing. Phys. Rev. A 83, 042317 (2011)

    Article  ADS  Google Scholar 

  15. Douglas, B.L., Wang, J.B.: A classical approach to the graph isomorphism problem using quantum walks. J. Phys. A 41, 075303 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  16. Carneiro, I., Loo, M., Xu, X., Girerd, M., Kendon, V., Knight, P.L.: Entanglement in coined quantum walks on regular graphs. New J. Phys. 7, 156 (2005)

    Article  ADS  Google Scholar 

  17. Montanaro, A.: Quantum walks on directed graphs. arXiv:quant-ph/0504116 (2005)

  18. Leung, G., Knott, P., Bailey, J., Kendon, V.: Coined quantum walks on percolation graphs. New J. Phys. 12, 123018 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  19. Stefanak, M., Bezdekova, I., Jex, I.: Limit distributions of three-state quantum walks: the role of coin eigenstates. Phys. Rev. A 90, 012342 (2014)

    Article  ADS  Google Scholar 

  20. Falkner, S., Boettcher, S.: Weak limit of the 3-state quantum walk on the line. Phys. Rev. A 90, 012307 (2014)

    Article  ADS  Google Scholar 

  21. Stefanak, M., Bezdekova, I., Jex, I.: Continuous deformations of the Grover walk preserving localization. Eur. Phys. J. D 66, 142 (2012)

    Article  ADS  Google Scholar 

  22. Stefanak, M., Bezdekova, I., Jex, I., Barnett, S.M.: Stability of point spectrum for three-state quantum walks on a line. Quant. Inf. Compu. 14(13-14), 1213–1226 (2014)

    MathSciNet  Google Scholar 

  23. Machida, T.: Limit theorems of a 3-state quantum walk and its application for discrete uniform measures. arXiv:1404.1522v1 (2014)

  24. Li, D., Mc Gettrick, M., Zhang, W.W., Zhang, K.J.: One dimensional lazy quantum walks and occupancy rate. arXiv:1412.6891 (2014)

  25. Delanty, M., Steel, M.J.: Discretely observable continuous-time quantum walks on Mobius strips and other exotic structures in three-dimensional integrated photonics. Phys. Rev. A 86, 043821 (2012)

    Article  ADS  Google Scholar 

  26. Jeffrey, M.P.: Mobius dual-mode resonators and bandpass filters. IEEE of Trans. Microwave Theory and Tech. 48(12), 2465C2471 (2000)

    MathSciNet  Google Scholar 

  27. Davis Richard, L.: Non-inductive electrical resistor. U.S. Patent 3,267,406 (1964)

  28. Enriquez, R.P.: A Structural parameter for High Tc Superconductivity from an Octahedral Moebius Strip in RBaCuO: 123 type of Perovskites. Rev. Mex. Fis. 48(supplement 1), 262 (2002). arXiv:cond-mat/0308019

    MathSciNet  Google Scholar 

  29. Alessandro, D.D., Parlangeli, G., Albertini, F.: Non-stationary quantum walks on the cycle. J. Phys. A: Math. Theor. 40, 14447–14455 (2007)

    Article  ADS  MATH  Google Scholar 

  30. Barr, K., Proctor, T., Hanson, B., Martiel, S., Pavlovic, V., Bullivant, A., Kendon, V.: Non-reversal and non-repeating quantum walks. Phys. Rev. A 89, 042332 (2014)

    Article  ADS  Google Scholar 

  31. Bach, E., Coppersmith, S., Goldschen, M.P., Joynt, R., Watrous, J.: One-dimensional quantum walks with absorbing boundaries. J. Comp. Syst. Sci. 69(4), 562C592 (2004)

    Article  MathSciNet  Google Scholar 

  32. Kwek, L.C., Setiawan: One-dimensional quantum walk with a moving boundary. Phys. Rev. A 84, 032319 (2011)

    Article  ADS  Google Scholar 

Download references

Acknowledgments

This work is supported by NSFC (Grant Nos. 61300181, 61272057, 61202434, 61170270, 61100203, 61121061), Beijing Natural Science Foundation (Grant No. 4122054), Beijing Higher Education Young Elite Teacher Project, BUPT Excellent Ph.D. Students Foundation(Grant Nos. CX201325, CX201326), China Scholarship Council(Grant Nos. 201306470046).

Author information

Authors and Affiliations

  1. State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Dan Li, Wei-Wei Zhang & Ke-Jia Zhang

  2. The De Brun Centre for Computational Algebra, School of Mathematics, The National University of Ireland, Galway, Ireland

    Dan Li & Michael Mc Gettrick

Authors
  1. Dan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Mc Gettrick
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei-Wei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ke-Jia Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dan Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, D., Mc Gettrick, M., Zhang, WW. et al. Quantum Walks on Two Kinds of Two-Dimensional Models. Int J Theor Phys 54, 2771–2783 (2015). https://doi.org/10.1007/s10773-015-2514-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-015-2514-5

Keywords

Search

Navigation