Quantum Information Processing

, Volume 14, Issue 3, pp 839–866 | Cite as

Quantum walk with a general coin: exact solution and asymptotic properties

Article

Abstract

In this paper, we present closed-form expressions for the wave function that governs the evolution of the discrete-time quantum walk on the line when the coin operator is arbitrary. The formulas were derived assuming that the walker can either remain put in the place or proceed in a fixed direction but never move backward, although they can be easily modified to describe the case in which the particle can travel in both directions. We use these expressions to explore properties of magnitudes associated to the process, as the probability mass function or the probability current, even though we also consider the asymptotic behavior of the exact solution. Within this approximation, we will estimate upper and lower bounds, examine the origins of an emerging approximate symmetry, and deduce the general form of the stationary probability density of the relative location of the walker.

Keywords

Quantum walks Closed-form solutions Limiting theorems 

References

  1. 1.
    Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687–1690 (1993)CrossRefADSGoogle Scholar
  2. 2.
    Nayak, N., Vishwanath, A.: Quantum Walk on the Line. arXiv:quant-ph/0010117 (2000)
  3. 3.
    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. ACM New York, New York, pp. 37–49 (2001)Google Scholar
  4. 4.
    Travaglione, B.C., Milburn, G.J.: Implementing the quantum random walk. Phys. Rev. A 65, 032310 (2002)CrossRefADSGoogle Scholar
  5. 5.
    Konno, N.: Quantum random walks in one dimension. Quantum Inf. Process. 1, 345–354 (2003)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44, 307–327 (2003)CrossRefADSGoogle Scholar
  7. 7.
    Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Inf. Process. 11, 1015–1106 (2012)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26, 1484–1509 (1997)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998)CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Childs, A., Farhi, E., Gutmann, S.: An example of the difference between quantum and classical random walks. Quantum Inf. Process. 1, 35–43 (2003)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Shenvi, N., Kempe, J., Whaley, K.B.: Quantum random-walk search algorithm. Phys. Rev. A 67, 052307 (2003)CrossRefADSGoogle Scholar
  12. 12.
    Agliari, E., Blumen, A., Nülken, O.: Quantum-walk approach to searching on fractal structures. Phys. Rev. A 82, 012305 (2010)CrossRefADSGoogle Scholar
  13. 13.
    Magniez, F., Nayak, A., Roland, J., Santha, M.: Search via quantum walk. SIAM J. Comput. 40, 142–164 (2011)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Flitney, A.P., Abbott, D., Johnson, N.F.: Quantum walks with history dependence. J. Phys. A 37, 7581–7591 (2004)CrossRefADSMATHMathSciNetGoogle Scholar
  15. 15.
    Bulger, D., Freckleton, J., Twamley, J.: Position-dependent and cooperative quantum Parrondo walks. New J. Phys. 10, 093014 (2008)CrossRefADSGoogle Scholar
  16. 16.
    Chandrashekar, C.M., Banerjee, S.: Parrondo’s game using a discrete-time quantum walk. Phys. Lett. A 375, 1553–1558 (2011)CrossRefADSMATHMathSciNetGoogle Scholar
  17. 17.
    Tregenna, B., Flanagan, W., Maile, R., Kendon, V.: Controlling discrete quantum walks: coins and initial states. New J. Phys. 5, 83 (2003)CrossRefADSGoogle Scholar
  18. 18.
    Bach, E., Coppersmith, S., Goldschen, M.P., Joynt, R., Watrous, J.: One-dimensional quantum walks with absorbing boundaries. J. Comput. Syst. Sci. 69, 562–592 (2004)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Chandrashekar, C.M., Srikanth, R., Laflamme, R.: Optimizing the discrete time quantum walk using a SU(2) coin. Phys. Rev. A 77, 032326 (2008)CrossRefADSGoogle Scholar
  20. 20.
    Chandrashekar, C.M., Srikanth, R., Banerjee, S.: Symmetries and noise in quantum walk. Phys. Rev. A 76, 022316 (2007)CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Asbóth, J.K.: Symmetries, topological phases, and bound states in the one-dimensional quantum walk. Phys. Rev. B 86, 195414 (2012)CrossRefADSGoogle Scholar
  22. 22.
    Kitagawa, T.: Topological phenomena in quantum walks: elementary introduction to the physics of topological phases. Quantum Inf. Process. 11, 1107–1148 (2012)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Fuss, I., White, L., Sherman, P., Naguleswaran, S.: An analytic solution for one-dimensional quantum walks. arXiv:0705.0077 (2007)
  24. 24.
    Villagra, M., Nakanishi, M., Yamashita, S., Nakashima, Y.: Quantum walks on the line with phase parameters. IEICE Trans. Inf. Syst. E95.D, 722–730 (2012)CrossRefADSGoogle Scholar
  25. 25.
    Grimmett, G., Janson, S., Scudo, P.F.: Weak limits for quantum random walks. Phys. Rev. E 69, 026119 (2004)CrossRefADSGoogle Scholar
  26. 26.
    Konno, N.: A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Jpn. 57, 1179–1195 (2005)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Bressler, A., Pemantle, R.: Quantum random walks in one dimension via generating functions. In: Proceedings of the 2007 Conference on Analysis of Algorithms, pp. 403–414 (2007)Google Scholar
  28. 28.
    Ahlbrecht, A., Vogts, H., Werner, A.H., Werner, R.F.: Asymptotic evolution of quantum walks with random coin. J. Math. Phys. 52, 042201 (2011)CrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Hoyer, S., Meyer, D.A.: Faster transport with a directed quantum walk. Phys. Rev. A 79, 024307 (2009)CrossRefADSGoogle Scholar
  30. 30.
    Montero, M.: Unidirectional quantum walks: evolution and exit times. Phys. Rev. A 88, 012333 (2013)CrossRefADSGoogle Scholar
  31. 31.
    Hillery, M., Bergou, J., Feldman, E.: Quantum walks based on an interferometric analogy. Phys. Rev. A 68, 032314 (2003)CrossRefADSMathSciNetGoogle Scholar
  32. 32.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics. Cambridge University Press, Cambridge (1953)Google Scholar
  33. 33.
    Romanelli, A.: Thermodynamic behavior of the quantum walk. Phys. Rev. A 85, 012319 (2012)CrossRefADSGoogle Scholar
  34. 34.
    Romanelli, A., Segundo, G.: The entanglement temperature of the generalized quantum walk. Phys. A 393, 646–654 (2014)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Departament de Física FonamentalUniversitat de Barcelona (UB)BarcelonaSpain

Personalised recommendations