Skip to main content
SpringerLink
Log in

Open Quantum Random Walks

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

A new model of quantum random walks is introduced, on lattices as well as on finite graphs. These quantum random walks take into account the behavior of open quantum systems. They are the exact quantum analogues of classical Markov chains. We explore the “quantum trajectory” point of view on these quantum random walks, that is, we show that measuring the position of the particle after each time-step gives rise to a classical Markov chain, on the lattice times the state space of the particle. This quantum trajectory is a simulation of the master equation of the quantum random walk. The physical pertinence of such quantum random walks and the way they can be concretely realized is discussed. Differences and connections with the already well-known quantum random walks, such as the Hadamard random walk, are established.

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

Similar content being viewed by others

References

  1. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1. Wiley, New York (1968)

    MATH  Google Scholar 

  2. Barber, M., Ninham, B.W.: Random and Restricted Walks: Theory and Applications. Gordon and Breach, New York (1970)

    MATH  Google Scholar 

  3. Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  4. Malkiel, B.: A Random Walk Down Wall Street. Norton, New York (1973)

    Google Scholar 

  5. Berg, H.C.: Random Walks in Biology. Princeton University Press, Princeton (1993)

    Google Scholar 

  6. Aharonov, Y., Davidovich, L., Zagury, N.: Phys. Rev. A 48, 1687 (1993)

    Article  ADS  Google Scholar 

  7. Farhi, E., Gutmann, S.: Phys. Rev. A 58, 915 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  8. Kempe, J.: Contemp. Phys. 44, 307 (2003)

    Article  ADS  Google Scholar 

  9. Konno, N.: A new type of limit theorems for one-dimensional quantum random walks. J. Math. Soc. Jpn. 57, 1179–1195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: In: Proceedings of the 33rd ACM Symposium on Theory of Computing, p. 50 (2001)

    Google Scholar 

  11. Venegas-Andraca, S.: Quantum Walks for Computer Scientists. Morgan and Claypool, San Rafael (2008)

    Google Scholar 

  12. Konno, N.: Quantum Walks in “Quantum Potential Theory”. Lecture Notes in Mathematics, p. 309. Springer, New York (2008)

    Book  Google Scholar 

  13. Ambainis, A.: Lect. Notes Comput. Sci. 4910, 1 (2008)

    Article  Google Scholar 

  14. Childs, A., Farhi, E., Gutmann, S.: Quantum Inf. Process. 1, 35 (2002)

    Article  MathSciNet  Google Scholar 

  15. Watrous, J.: J. Comput. Syst. Sci. 62, 376 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Childs, A., Cleve, R., Deotto, E., Farhi, E.: In: Proceedings of the 35th ACM Symposium on Theory of Computing, p. 59 (2003)

    Google Scholar 

  17. Shenvi, N., Kempe, J., Whaley, K.B.: Phys. Rev. A 67, 052307 (2003)

    Article  ADS  Google Scholar 

  18. Farhi, E., Goldstone, J., Gutmann, S.: Theory Comput. 4, 169 (2008)

    Article  MathSciNet  Google Scholar 

  19. Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, London (2002)

    MATH  Google Scholar 

  20. Karski, M., Forster, L., Choi, J.-M., Steffen, A., Alt, W., Meschede, D., Widera, A.: Science 325, 174 (2009)

    Article  ADS  Google Scholar 

  21. Perets, H.B., Lahini, Y., Pozzi, F., Sorel, M., Morandotti, R., Silberberg, Y.: Phys. Rev. Lett. 100, 170506 (2008)

    Article  ADS  Google Scholar 

  22. Schmitz, H., Matjeschk, R., Schneider, Ch., Glueckert, J., Enderlein, M., Huber, T., Schaetz, T.: Phys. Rev. Lett. 103, 090504 (2009)

    Article  ADS  Google Scholar 

  23. Zähringer, F., Kirchmair, G., Gerritsma, R., Solano, E., Blatt, R., Roos, C.F.: Phys. Rev. Lett. 104, 100503 (2010)

    Article  Google Scholar 

  24. Broome, M.A., Fedrizzi, A., Lanyon, B.P., Kassal, I., Aspuru-Guzik, A., White, A.G.: Phys. Rev. Lett. 104, 153602 (2010)

    Article  ADS  Google Scholar 

  25. Rebentrost, P., Mohseni, M., Aspuru-Guzik, A.: J. Phys. Chem. B 113, 9942 (2009)

    Article  Google Scholar 

  26. Mohseni, M., Rebentrost, P., Lloyd, S., Aspuru-Guzik, A.: J. Chem. Phys. 129, 174106 (2008)

    Article  ADS  Google Scholar 

  27. Plenio, M., Huelga, S.: New J. Phys. 10, 113019 (2008)

    Article  ADS  Google Scholar 

  28. Caruso, F., Chin, A., Datta, A., Huelga, S., Plenio, M.: J. Chem. Phys. 131, 105106 (2009)

    Article  ADS  Google Scholar 

  29. Rebentrost, P., Mohseni, M., Kassal, I., Lloyd, S., Aspuru-Guzik, A.: New J. Phys. 11, 033003 (2009)

    Article  ADS  Google Scholar 

  30. Kendon, V.: Math. Struct. Comput. Sci. 17, 1169 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Brun, T.A., Carteret, H.A., Ambainis, A.: Phys. Rev. Lett. 91, 130602 (2003)

    Article  ADS  Google Scholar 

  32. Romanelli, A., Siri, R., Abal, G., Auyuanet, A., Donangelo, R.: Physica A 347, 137 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  33. Love, P., Boghosian, B.: Quantum Inf. Process. 4, 335 (2005)

    Article  MATH  Google Scholar 

  34. Srikanth, R., Banerjee, S., Chandrashekar, C.M.: Phys. Rev. A 81, 062123 (2010)

    Article  ADS  Google Scholar 

  35. Kendon, V., Tregenna, B.: Phys. Rev. A 67, 042315 (2003)

    Article  ADS  Google Scholar 

  36. Whitfield, J.D., Rodríguez-Rosario, C.A., Aspuru-Guzik, A.: Phys. Rev. A 81, 022323 (2010)

    Article  ADS  Google Scholar 

  37. Pandey, D., Satapathy, N., Meena, M.S., Ramachandran, H.: Phys. Rev. A 84, 042322 (2011)

    Article  ADS  Google Scholar 

  38. Broome, M.A., et al.: Phys. Rev. Lett. 104, 153602 (2010)

    Article  ADS  Google Scholar 

  39. Schreiber, A., et al.: Phys. Rev. Lett. 106, 180403 (2011)

    Article  ADS  Google Scholar 

  40. Regensburger, A., et al.: Phys. Rev. Lett. 107, 233902 (2011)

    Article  ADS  Google Scholar 

  41. Ahlbrecht, A., Vogts, H., Werner, A.H., Werner, R.F.: J. Math. Phys. 52, 042201 (2011)

    Article  ADS  Google Scholar 

  42. Gudder, S.: Found. Phys. 40(9–10), 1566 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. Gudder, S.: J. Math. Phys. 49, 072105 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  44. Kraus, K.: States, Effects and Operations: Fundamental Notions of Quantum Theory. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  45. Alicki, R., Lendi, K.: Quantum Dynamical Semigroups and Applications. Springer, Berlin (1987)

    MATH  Google Scholar 

  46. Attal, S., Guillotin-Plantard, N., Sabot, C.: Central-Limit Theorems for open quantum random walks. Preprint

  47. Renger, T., May, V.: J. Phys. Chem. A 102, 4381 (1998)

    Article  Google Scholar 

  48. Attal, S., Pautrat, Y.: From repeated to continuous quantum interactions. Ann. Inst. Henri Poincaré. Phys. Théor. 7, 59–104 (2006)

    MathSciNet  ADS  MATH  Google Scholar 

  49. Haroche, S., Gleyzes, S., Kuhr, S., Guerlin, C., Bernu, J., Deléglise, S., Busk-Hoff, U., Brune, M., Raimond, J.-M.: Nature 446, 297 (2007)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Camille Jordan, Université de Lyon, Université de Lyon 1, C.N.R.S., 21 av Claude Bernard, 69622, Villeubanne cedex, France

    S. Attal & C. Sabot

  2. Quantum Research Group, School of Physics and National Institute for Theoretical Physics, University of KwaZulu-Natal, Durban, 4001, South Africa

    F. Petruccione & I. Sinayskiy

Authors
  1. S. Attal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Petruccione
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Sabot
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. I. Sinayskiy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Attal.

Additional information

Work supported by ANR project “HAM-MARK”, N° ANR-09-BLAN-0098-01, by South African Research Chair Initiative of the Department of Science and Technology and National Research Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Attal, S., Petruccione, F., Sabot, C. et al. Open Quantum Random Walks. J Stat Phys 147, 832–852 (2012). https://doi.org/10.1007/s10955-012-0491-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-012-0491-0

Keywords

Search

Navigation