Advertisement

Quantum Information Processing

, Volume 14, Issue 9, pp 3179–3191 | Cite as

Moments of coinless quantum walks on lattices

  • Raqueline Azevedo Medeiros SantosEmail author
  • Renato Portugal
  • Stefan Boettcher
Article

Abstract

The properties of the coinless quantum-walk model have not been as thoroughly analyzed as those of the coined model. Both evolve in discrete time steps, but the former uses a smaller Hilbert space, which is spanned merely by the site basis. Besides, the evolution operator can be obtained using a process of lattice tessellation, which is very appealing. The moments of the probability distribution play an important role in the context of quantum walks. The ballistic behavior of the mean square displacement indicates that quantum-walk-based algorithms are faster than random-walk-based ones. In this paper, we obtain analytical expressions for the moments of the coinless model on d-dimensional lattices by employing the methods of Fourier transforms and generating functions. The mean square displacement for large times is explicitly calculated for the one- and two-dimensional lattices, and using optimization methods, the parameter values that give the largest spread are calculated and compared with the equivalent ones of the coined model. Although we have employed asymptotic methods, our approximations are accurate even for small numbers of time steps.

Keywords

Coinless quantum walks Moments Mean square displacement Standard deviation 

Notes

Acknowledgments

RAMS acknowledges financial support from Capes-Faperj E-45/2013. RP thanks Faperj (Grant No. E-26/102.350/2013) and CNPq (Grant Nos. 304709/2011-5, 474143/2013-9, and 400216/2014-0). SB acknowledges financial support from the US National Science Foundation through Grant DMR-1207431.

References

  1. 1.
    Portugal, R.: Quantum Walks and Search Algorithms. Springer, New York (2013)zbMATHCrossRefGoogle Scholar
  2. 2.
    Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48(2), 1687–1690 (1993)CrossRefADSGoogle Scholar
  3. 3.
    Patel, A., Raghunathan, K.S., Rungta, P.: Quantum random walks do not need a coin toss. Phys. Rev. A 71, 032347 (2005)MathSciNetCrossRefADSGoogle Scholar
  4. 4.
    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, NY, USA (2001)Google Scholar
  5. 5.
    Nayak, A., Vishwanath, A.: Quantum Walk on the Line (2000). arXiv:quant-ph/0010117v1
  6. 6.
    Konno, N.: Quantum random walks in one dimension. Quantum Inform. Process. 1(5), 345–354 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Grimmett, G., Janson, S., Scudo, P.F.: Weak limits for quantum random walks. Phys. Rev. E 69, 026119 (2004)CrossRefADSGoogle Scholar
  8. 8.
    Shikano, Y.: From discrete time quantum walk to continuous time quantum walk in limit distribution. J. Comput. Theor. Nanosci. 10(7), 1558–1570 (2013)CrossRefGoogle Scholar
  9. 9.
    Štefaňák, M., Bezděková, I., Jex, I.: Limit distributions of three-state quantum walks: the role of coin eigenstates. Phys. Rev. A 90, 012342 (2014)CrossRefGoogle Scholar
  10. 10.
    Falkner, S., Boettcher, S.: Weak limit of the three-state quantum walk on the line. Phys. Rev. A 90, 012307 (2014)CrossRefADSGoogle Scholar
  11. 11.
    Watabe, K., Kobayashi, N., Katori, M., Konno, N.: Limit distributions of two-dimensional quantum walks. Phys. Rev. A 77, 062331 (2008)CrossRefADSGoogle Scholar
  12. 12.
    Ampadu, C.: Limit theorems for quantum walks associated with Hadamard matrices. Phys. Rev. A 84, 012324 (2011)CrossRefADSGoogle Scholar
  13. 13.
    Proctor, T.J., Barr, K.E., Hanson, B., Martiel, S., Pavlović, V., Bullivant, A., Kendon, V.M.: Nonreversal and nonrepeating quantum walks. Phys. Rev. A 89, 042332 (2014)CrossRefADSGoogle Scholar
  14. 14.
    Falk, M.: Quantum search on the spatial grid (2013). arXiv:quant-ph/1303.4127
  15. 15.
    Patel, A., Raghunathan, K.S., Rahaman, M.A.: Search on a hypercubic lattice using a quantum random walk. ii. \(d=2\). Phys. Rev. A 82, 032331 (2010)MathSciNetCrossRefADSGoogle Scholar
  16. 16.
    Ambainis, A., Portugal, R., Nahimov, N.: Spatial search on grids with minimum memory. arXiv:quant-ph/1312.0172
  17. 17.
    Portugal, R., Boettcher, S., Falkner, S.: One-dimensional coinless quantum walks (2014). arXiv:1408.5166v2 [quant-ph]
  18. 18.
    Tregenna, B., Flanagan, W., Maile, R., Kendon, V.: Controlling discrete quantum walks: coins and initial states. New J. Phys. 5(1), 83 (2003)CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Raqueline Azevedo Medeiros Santos
    • 1
    Email author
  • Renato Portugal
    • 1
    • 2
  • Stefan Boettcher
    • 3
  1. 1.Laboratório Nacional de Computação CientíficaPetrópolisBrazil
  2. 2.Universidade Católica de PetrópolisPetrópolisBrazil
  3. 3.Department of PhysicsEmory UniversityAtlantaUSA

Personalised recommendations