Skip to main content
SpringerLink
Log in

Asymptotic distributions of quantum walks on the line with two entangled coins

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

We advance the previous studies of quantum walks on the line with two coins. Such four-state quantum walks driven by a three-direction shift operator may have nonzero limiting probabilities (localization), thereby distinguishing them from the quantum walks on the line in the basic scenario (i.e., driven by a single coin). In this work, asymptotic position distributions of the quantum walks are examined. We derive a weak limit for the quantum walks and explicit formulas for the limiting probability distribution, whose dependencies on the coin parameter and the initial state of quantum walks are presented. In particular, it is shown that the weak limit for the present quantum walks can be of the form in the basic scenario of quantum walks on the line, for certain initial states of the walk and certain values of the coin parameter. In the case where localization occurs, we show that the limiting probability decays exponentially in the absolute value of a walker’s position, independent of the parity of time.

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

Similar content being viewed by others

References

  1. Nayak, A., Vishwanath, A.: Quantum Walk on the Line, arXiv:quantph/0010117. (2000)

  2. Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, STOC ’01, pp. 37–49. ACM, New York (2001)

  3. Aharanov, 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. ACM, New York (2001)

  4. Farhi E., Gutmann S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  5. Childs A., Farhi E., Gutmann S.: An example of a difference between quantum and classical random walks. Quant. Inf. Proc. 1, 35 (2002)

    Article  MathSciNet  Google Scholar 

  6. Kempe J.: Quantum random walks—an introductory overview. Contemp. Phys. 44, 307 (2003)

    Article  ADS  Google Scholar 

  7. Konno, N.: Quantum Potential Theory. Lecture Notes in Mathematics, In: Franz, U., Schurmann, M. (ed.) pp. 309–452. Springer, Heidelberg (2008)

  8. Venegas-Andraca, S.E.: Quantum Walks for Computer Scientists. Morgan and Claypool Publishers (Synthesis Lectures on Quantum Computing), 2008

  9. Konno N.: Quantum random walks in one dimension. Quantum Inf. Proc. 1, 345–354 (2002)

    Article  MathSciNet  Google Scholar 

  10. Konno N.: A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Jap. 57, 1179–1195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grimmett G., Janson S., Scudo P.F.: Weak limits for quantum random walks. Phys. Rev. E 69, 026119 (2004)

    Article  ADS  Google Scholar 

  12. Venegas-Andraca S.E., Ball J.L., Burnett K., Bose S.: Quantum walks with entangled coins. New J. Phys. 7, 221 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  13. Liu C., Petulante N.: One-dimensional quantum random walk with two entangled coins. Phys. Rev. A 79, 032312 (2009)

    Article  ADS  Google Scholar 

  14. Inui N., Konno N., Segawa E.: One-dimensional three-state quantum walk. Phys. Rev. E 72, 056112 (2005)

    Article  ADS  Google Scholar 

  15. Konno N., Machida T.: Limit theorems for quantum walks with memory. Quant. Inf. Comput. 10(11&12), 1004–1017 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Mc Gettrick M.: One dimensional quantum walks with memory. Quant. Inf. Comput. 10, 509–524 (2010)

    MathSciNet  Google Scholar 

  17. Brun T.A., Carteret H.A., Ambainis A.: Quantum walks driven by many coins. Phys. Rev. A 67, 052317 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  18. Segawa E., Konno N.: Limit theorems for quantum walks driven by many coins. Int. J. Quant. Inf. 6, 1231–1243 (2008)

    Article  MATH  Google Scholar 

  19. Ampadu C.: Limit theorems for quantum walks associated with Hadamard matrices. Phys. Rev. A 84, 012324 (2011)

    Article  ADS  Google Scholar 

  20. Cantero M.J., Grünbaum F.A., Moral L., Velázquez L.: Matrix-valued Szegö polynomials and quantum random walks. Commun. Pure Appl. Math. 63, 464–507 (2010)

    MATH  Google Scholar 

  21. Konno N., Segawa E.: Localization of discrete-time quantum walks on a half line via the CGMV method. Quant. Inf. Comput. 11, 0485–0495 (2011)

    MathSciNet  Google Scholar 

  22. Ahlbrecht A., Vogts H., Werner A.H., Werner R.F.: Asymptotic evolution of quantum walks with random coin. J. Math. Phys. 52, 042201 (2011)

    Article  ADS  Google Scholar 

  23. Fan S., Feng Z., Xiong S., Yang W.S.: Convergence of quantum random walks with decoherence. Phys. Rev. A 84, 042317 (2011)

    Article  ADS  Google Scholar 

  24. Yang W.S., Liu C., Zhang K.: A path integral formula with applications to quantum random walks in \({\mathbb{Z}^d}\) . J. Phys. A Math. Theor. 40, 8487–8516 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Liu C.: The “t/2 law” for quantum random walks on the line starting in the classical state. J. Phys. A Math. Theor. 41, 355306 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Bowie State University, Bowie, MD, 20715, USA

    Chaobin Liu

Authors
  1. Chaobin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chaobin Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, C. Asymptotic distributions of quantum walks on the line with two entangled coins. Quantum Inf Process 11, 1193–1205 (2012). https://doi.org/10.1007/s11128-012-0361-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-012-0361-3

Keywords

Search

Navigation