Advertisement

Discrete time quantum walks on percolation graphs

  • Bálint Kollár
  • Jaroslav Novotný
  • Tamás Kiss
  • Igor Jex
Open Access
Regular Article
Part of the following topical collections:
  1. Focus Point on Quantum information and complexity

Abstract.

Randomly breaking connections in a graph alters its transport properties, a model used to describe percolation. In the case of quantum walks, dynamic percolation graphs represent a special type of imperfections, where the connections appear and disappear randomly in each step during the time evolution. The resulting open system dynamics is hard to treat numerically in general. We shortly review the literature on this problem. We then present our method to solve the evolution on finite percolation graphs in the long time limit, applying the asymptotic methods concerning random unitary maps. We work out the case of one-dimensional chains in detail and provide a concrete, step-by-step numerical example in order to give more insight into the possible asymptotic behavior. The results about the case of the two-dimensional integer lattice are summarized, focusing on the Grover-type coin operator.

Keywords

Edge State Quantum Walk Shift Condition Position Distribution Break Link 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Y. Aharonov, L. Davidovich, N. Zagury, Phys. Rev. A 48, 1687 (1993)CrossRefADSGoogle Scholar
  2. 2.
    David A. Meyer, J. Stat. Phys. 85, 551 (1996)CrossRefzbMATHADSGoogle Scholar
  3. 3.
    O. Mülken, V. Pernice, A. Blumen, Phys. Rev. E 76, 051125 (2007)CrossRefADSGoogle Scholar
  4. 4.
    N. Inui, Y. Konishi, N. Konno, Phys. Rev. A 69, 052323 (2004)CrossRefADSGoogle Scholar
  5. 5.
    M. Stefaňák, I. Jex, T. Kiss, Phys. Rev. Lett. 100, 020501 (2008)CrossRefADSGoogle Scholar
  6. 6.
    Y. Shikano, H. Katsura, Phys. Rev. E 82, 031122 (2010)MathSciNetCrossRefADSGoogle Scholar
  7. 7.
    G.D. Paparo, M.A. Martin-Delgado, Nature Sci. Rep. 2, 444 (2012)ADSGoogle Scholar
  8. 8.
    J. Asbóth, H. Obuse, Phys. Rev. B 88, 121406 (2013)CrossRefADSGoogle Scholar
  9. 9.
    A.M. Childs, Phys. Rev. Lett. 102, 180501 (2009)MathSciNetCrossRefADSGoogle Scholar
  10. 10.
    N.B. Lovett, S. Cooper, M. Everitt, M. Trevers, V. Kendon, Phys. Rev. A 81, 042330 (2010)MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    J. Kempe, Cont. Phys. 44, 307 (2003)CrossRefADSGoogle Scholar
  12. 12.
    N. Konno, Quantum walks, in Quantum Potential Theory, edited by U. Franz, M. Schürmann (Springer, 2008)Google Scholar
  13. 13.
    M. Santha, Quantum walk based search algorithms, in Theory and Applications of Models of Computation, edited by Manindra Agrawal, Dingzhu Du, Zhenhua Duan, Angsheng Li, Lecture Notes in Computer Science, Vol. 4978 (Springer, Berlin-Heidelberg, 2008)Google Scholar
  14. 14.
    S.E. Venegas-Andraca, Quant. Inf. Proc. 11, 1015 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, A. Widera, Science 325, 174 (2009)CrossRefADSGoogle Scholar
  16. 16.
    H. Schmitz, R. Matjeschk, Ch. Schneider, J. Glueckert, M. Enderlein, T. Huber, T. Schaetz, Phys. Rev. Lett. 103, 090504 (2009)CrossRefADSGoogle Scholar
  17. 17.
    F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, C.F. Roos, Phys. Rev. Lett. 104, 100503 (2010)CrossRefGoogle Scholar
  18. 18.
    A. Peruzzo, M. Lobino, J.C.F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M.G. Thompson, J.L. O'Brien, Science 329, 1500 (2010)CrossRefADSGoogle Scholar
  19. 19.
    M.A. Broome, A. Fedrizzi, B.P. Lanyon, I. Kassal, A. Aspuru-Guzik, A.G. White, Phys. Rev. Lett. 104, 153602 (2010)CrossRefADSGoogle Scholar
  20. 20.
    A. Schreiber, K.N. Cassemiro, V. Potoček, A. Gábris, P.J. Mosley, E. Andersson, I. Jex, Ch. Silberhorn, Phys. Rev. Lett. 104, 050502 (2010)CrossRefADSGoogle Scholar
  21. 21.
    A. Schreiber, K.N. Cassemiro, V. Potoček, A. Gábris, Abris, I. Jex, Ch. Silberhorn, Phys. Rev. Lett. 106, 180403 (2011)CrossRefADSGoogle Scholar
  22. 22.
    A. Schreiber, A. Gábris, P.P. Rohde, K. Laiho, M. Stefaňák, V. Potoček, C. Hamilton, I. Jex, Ch. Silberhorn, Science 336, 55 (2012)CrossRefADSGoogle Scholar
  23. 23.
    L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, R. Osellame, Phys. Rev. Lett. 108, 010502 (2012)CrossRefADSGoogle Scholar
  24. 24.
    J. Svozilík, R. de J. León-Montiel, J.P. Torres, Phys. Rev. A 86, 052327 (2012)CrossRefADSGoogle Scholar
  25. 25.
    Y. Lahini, M. Verbin, S.D. Huber, Y. Bromberg, R. Pugatch, Y. Silberberg, Phys. Rev. A 86, 011603 (2012)CrossRefADSGoogle Scholar
  26. 26.
    M. Genske, W. Alt, A. Steffen, A.H. Werner, R.F. Werner, D. Meschede, A. Alberti, Phys. Rev. Lett. 110, 190601 (2013)CrossRefADSGoogle Scholar
  27. 27.
    A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, P. Mataloni, Nature Photonics 7, 322 (2013)CrossRefADSGoogle Scholar
  28. 28.
    J.D.A. Meinecke, K. Poulios, A. Politi, J.C.F Matthews, A. Peruzzo, N. Ismail, K. Wörhoff, J.L. O'Brien, M.G. Thompson, Phys. Rev. A 88, 012308 (2013)CrossRefADSGoogle Scholar
  29. 29.
    M. Hillery, J. Bergou, E. Feldman, Phys. Rev. A 68, 032314 (2003)MathSciNetCrossRefADSGoogle Scholar
  30. 30.
    E. Farhi, S. Gutmann, Phys. Rev. A 58, 915 (1998)MathSciNetCrossRefADSGoogle Scholar
  31. 31.
    M. Szegedy, Quantum speed-up of markov chain based algorithms, in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (IEEE, 2004)Google Scholar
  32. 32.
    V. Kendon, Math. Struct. Comput. Sci. 17, 1169 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    I. Sinayskiy, F. Petruccione, J. Phys. Conf. Ser. 442, 012003 (2013)CrossRefADSGoogle Scholar
  34. 34.
    N. Konno, H. Yoo, J. Stat. Phys. 150, 299 (2013)MathSciNetCrossRefzbMATHADSGoogle Scholar
  35. 35.
    A. Ahlbrecht, H. Vogts, A.H. Werner, R.F. Werner, J. Math. Phys. 52, 042201 (2011)MathSciNetCrossRefADSGoogle Scholar
  36. 36.
    H. Kesten, Percolation theory for mathematicians (Birkhäuser Boston, 1982)Google Scholar
  37. 37.
    G. Grimmett, Percolation, in Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen (Springer, 1999)Google Scholar
  38. 38.
    G. Leung, P. Knott, J. Bailey, V. Kendon, New J. Phys. 12, 123018 (2010)MathSciNetCrossRefADSGoogle Scholar
  39. 39.
    J.E. Steif, A survey of dynamical percolation, in Fractal geometry and stochastics IV, edited by C. Bandt, M. Zähle, P. Mörters (Springer, 2009)Google Scholar
  40. 40.
    A. Romanelli, R. Siri, G. Abal, A. Auyuanet, R. Donangelo, Phys. A: Stat. Mech. Appl. 347, 137 (2005)MathSciNetCrossRefGoogle Scholar
  41. 41.
    A. Romanelli, A. Hernández, Phys. A: Stat. Mech. Appl. 390, 1209 (2011)CrossRefGoogle Scholar
  42. 42.
    A.C. Oliveira, R. Portugal, R. Donangelo, Phys. Rev. A 74, 012312 (2006)CrossRefADSGoogle Scholar
  43. 43.
    G. Abal, R. Donangelo, F. Severo, R. Siri, Phys. A: Stat. Mech. Appl. 387, 335 (2008)MathSciNetCrossRefGoogle Scholar
  44. 44.
    M. Annabestani, S.J. Akhtarshenas, M.R. Abolhassani, Phys. Rev. A 81, 032321 (2010)CrossRefADSGoogle Scholar
  45. 45.
    F.L. Marquezino, R. Portugal, G. Abal, R. Donangelo, Phys. Rev. A 77, 042312 (2008)MathSciNetCrossRefADSGoogle Scholar
  46. 46.
    R.A. Santos, R. Portugal, M.D. Fragoso, Quantum Inf. Process. 13, 559 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    B. Kollár, T. Kiss, J. Novotný, I. Jex, Phys. Rev. Lett. 108, 230505 (2012)CrossRefADSGoogle Scholar
  48. 48.
    B. Kollár, J. Novotný, T. Kiss, I. Jex, New J. Phys. 16, 023002 (2014)CrossRefADSGoogle Scholar
  49. 49.
    M. Hinarejos, C. Di Franco, A. Romanelli, A. Pérez, Chirality asymptotic behavior and non-markovianity in quantum walks on a line, arXiv:1401.3243
  50. 50.
    C. Ampadu, Commun. Theor. Phys. 57, 41 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    N.B. Lovett, M. Everitt, R.M. Heath, V. Kendon, The quantum walk search algorithm: Factors affecting efficiency, arXiv:1110.4366
  52. 52.
    C.M. Chandrashekar, T. Busch, Quantum percolation and anderson transition point for transport of a two-state particle, arXiv:1303.7013
  53. 53.
    C.M. Chandrashekar, S. Melville, Th. Busch, J. Phys. B: At. Mol. Opt. Phys. 47, 085502 (2014)CrossRefADSGoogle Scholar
  54. 54.
    K.R. Motes, A. Gilchrist, P.P. Rohde, Quantum random walks on congested lattices, arXiv:1310.8161
  55. 55.
    O. Mülken, A. Blumen, Phys. E: Low-dimensional Syst. Nanostruct. 42, 576 (2010)CrossRefADSGoogle Scholar
  56. 56.
    O. Mülken, Olken, A. Blumen, Phys. Rep. 502, 37 (2011)MathSciNetCrossRefADSGoogle Scholar
  57. 57.
    A. Anishchenko, A. Blumen, O. Mülken, Quantum Inf. Process 11, 1273 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Z. Darázs, T. Kiss, J. Phys. A: Math. Theor. 46, 375305 (2013)CrossRefGoogle Scholar
  59. 59.
    J. Novotný, G. Alber, I. Jex, J. Phys. A: Math. Theor. 42, 282003 (2009)CrossRefGoogle Scholar
  60. 60.
    J. Novotný, G. Alber, I. Jex, Centr. Eur. J. Phys. 8, 1001 (2010)CrossRefADSGoogle Scholar
  61. 61.
    J. Novotný, G. Alber, I. Jex, J. Phys. A: Math. Theor. 45, 485301 (2012)CrossRefGoogle Scholar
  62. 62.
    T.D. Mackay, S.D. Bartlett, L.T. Stephenson, B.C. Sanders, J. Phys. A 35, 2745 (2002)MathSciNetCrossRefzbMATHADSGoogle Scholar
  63. 63.
    B. Tregenna, W. Flanagan, R. Maile, V. Kendon, New J. Phys. 5, 83 (2003)CrossRefADSGoogle Scholar
  64. 64.
    N. Shenvi, J. Kempe, K.B. Whaley, Phys. Rev. A 67, 052307 (2003)CrossRefADSGoogle Scholar
  65. 65.
    M. Stefaňák, B. Kollár, T. Kiss, I. Jex, Phys. Scr. T140, 014035 (2010)CrossRefADSGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Bálint Kollár
    • 1
  • Jaroslav Novotný
    • 2
  • Tamás Kiss
    • 1
  • Igor Jex
    • 2
  1. 1.Wigner RCP, SZFKIBudapestHungary
  2. 2.Department of Physics, Faculty of Nuclear Sciences and Physical EngineeringCzech Technical University in PraguePraha 1 - Staré MěstoCzech Republic

Personalised recommendations