Quantum Information Processing

, Volume 11, Issue 5, pp 1219–1249 | Cite as

Asymptotic behavior of quantum walks with spatio-temporal coin fluctuations

  • Andre Ahlbrecht
  • Christopher Cedzich
  • Robert Matjeschk
  • Volkher B. Scholz
  • Albert H. Werner
  • Reinhard F. Werner


Quantum walks subject to decoherence generically suffer the loss of their genuine quantum feature, a quadratically faster spreading compared to classical random walks. This intuitive statement has been verified analytically for certain models and is also supported by numerical studies of a variety of examples. In this paper we analyze the long-time behavior of a particular class of decoherent quantum walks, which, to the best of our knowledge, was only studied at the level of numerical simulations before. We consider a local coin operation which is randomly and independently chosen for each time step and each lattice site and prove that, under rather mild conditions, this leads to classical behavior: With the same scaling as needed for a classical diffusion the position distribution converges to a Gaussian, which is independent of the initial state. Our method is based on non-degenerate perturbation theory and yields an explicit expression for the covariance matrix of the asymptotic Gaussian in terms of the randomness parameters.


Quantum walk Spatio-temporal coin fluctuation Asymptotic behavior Perturbation theory 


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  1. 1.
    Abal G., Donangelo R., Severo F., Siri R.: Decoherent quantum walks driven by a generic coin operation. Phys. A Stat. Mech. App. 387, 335–345 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ahlbrecht, A., Scholz,V., Werner, A. : Disordered quantum walks in one lattice dimension. J. Math. Phys. 52, 102, 201 (2011). doi: 10.1063/1.3643768 Google Scholar
  3. 3.
    Ahlbrecht, A., Vogts, H., Werner, A., Werner, R.: Asymptotic evolution of quantum walks with random coin. J. Math. Phys. 52, 042, 201 (2011). doi: 10.1063/1.3575568 Google Scholar
  4. 4.
    Ambainis A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 1, 507 (2003)zbMATHCrossRefGoogle Scholar
  5. 5.
    Ambainis A.: Quantum walk algorithm for element distinctness. SIAM J. Comput. 37, 210–239 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Banuls, M.C., Navarrete, C., Pérez, A., Roldán, E., Soriano, J.C.: Quantum walk with a time-dependent coin. Phys. Rev. A 73, 062, 304 (2006). doi: 10.1103/PhysRevA.73.062304
  7. 7.
    Brun, T.A., Carteret, H.A., Ambainis, A.: Quantum walks driven by many coins. Phys. Rev. A 67, 052, 317 (2002)Google Scholar
  8. 8.
    Chandrashekar, C., Srikanth, R., Banerjee, S.: Symmetries and noise in quantum walk. Phys. Rev. A 76, 022, 316 (2007)Google Scholar
  9. 9.
    Childs A.M., Cleve R., Jordan S.P., Yeung D.: Discrete-query quantum algorithm for nand trees. Theory Comput. 5, 119–123 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Farhi E., Goldstone J., Gutmann S.: A quantum algorithm for the hamiltonian nand tree. Theory Comput. 4, 169–190 (2008). doi: 10.4086/toc.2008.v004a008 MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gross D., Nesme V., Vogts H., Werner R.F.: Index theory of one dimensional quantum walks and cellular automata. Comm. Math. Phys. 310(2), 419–454 (2012). doi: 10.1007/s00220-012-1423-1 MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Hamza, E., Joye, A.: Correlated markov quantum walks (2011). arXiv:1110.4862Google Scholar
  13. 13.
    Joye A.: Random time-dependent quantum walks. Comm. Math. Phys. 307, 65–100 (2011). doi: 10.1007/s00220-011-1297-7 MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Joye A., Merkli M.: Dynamical localization of quantum walks in random environments. J. Stat. Phys. 140, 1–29 (2010). doi: 10.1007/s10955-010-0047-0 MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Karski M., Förster L., Choi J.M., Steffen A., Alt W., Meschede D., Widera A.: Quantum walk in position space with single optically trapped atoms. Science 325, 174 (2009). doi: 10.1126/science.1174436 ADSCrossRefGoogle Scholar
  16. 16.
    Kato T.: Perturbation Theory for Linear Operators. Springer, New York, NY (1995)zbMATHGoogle Scholar
  17. 17.
    Kempe J.: Quantum random walks hit exponentially faster. Probab. Theory Rel. 133(2), 215–235 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Konno N.: Localization of an inhomogeneous discrete-time quantum walk on the line. Quantum Inf. Process. 9(3), 405–418 (2010). doi: 10.1007/s11128-009-0147-4 MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Konno N.: One-dimensional discrete-time quantum walks on random environments. Quantum Inf. Proc. 8(5), 387–399 (2009). doi: 10.1007/s11128-009-0116-y MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. 20.
    Košík, J., Bužek, V., Hillery, M.: Quantum walks with random phase shifts. Phys. Rev. A 74, 022, 310 (2006)Google Scholar
  21. 21.
    Leung G., Knott P., Bailey J., Kendon V.: Coined quantum walks on percolation graphs. New J. Phys. 12, 1–24 (2010). doi: 10.1088/1367-2630/12/12/123018 MathSciNetCrossRefGoogle Scholar
  22. 22.
    Matjeschk, R., Schneider, C., Enderlein, M., Huber, T., Schmitz, H., Glueckert, J., Schaetz, T.: Experimental simulation and limitations of quantum walks with trapped ions. New J. Phys. (2011). arXiv:1108.0913Google Scholar
  23. 23.
    Navarette-Benlloch, C., Pérez, A., Roldán, E.: Nonlinear optical Galton board. Phys. Rev. A 75, 062, 333 (2007)Google Scholar
  24. 24.
    Negele, J., Orland, H.: Quantum many-particle systems. Advanced Books Classics. Perseus Books (1998)Google Scholar
  25. 25.
    Obuse H., Kawakami N.: Topological phases and delocalization of quantum walks in random environments. Phys. Rev. B 84(19), 1–6 (2011). doi: 10.1103/PhysRevB.84.195139 CrossRefGoogle Scholar
  26. 26.
    Perez, A., Romanelli, A.: Effects of broken links on the long-time behavior of quantum walks (2011). arXiv:1109.0122Google Scholar
  27. 27.
    Reed M., Simon B.: Methods of Modern Mathematical Physics, vol IV Analysis of Operators. Academic Press, New York, NY (1978)Google Scholar
  28. 28.
    Ribeiro, P., Milman, P., Mosseri, R.: Aperiodic quantum random walks. Phys. Rev. Lett. 93(19), 190, 503 (2004). doi: 10.1103/PhysRevLett.93.190503
  29. 29.
    Romanelli, A., Auyuanet, A., Siri, R., Abal, G., Donangelo, R.: Generalized quantum walk in momentum space. Phys. A 352, 409 (2005)Google Scholar
  30. 30.
    Romanelli A., Siri R., Abal G., Auyuanet A., Donangelo R.: Decoherence in the quantum walk on the line. Phys. A Stat. Mech. App. 347, 137–152 (2005)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Schmitz, H., Matjeschk, R., Schneider, C., Glueckert, J., Enderlein, M., Huber, T., Schaetz, T.: Quantum walk of a trapped ion in phase space. Phys. Rev. Lett. 103(9), 090, 504 (2009). doi: 10.1103/PhysRevLett.103.090504
  32. 32.
    Schreiber, A., Cassemiro, K.N., Potoček, V., Gábris, A., Mosley, P.J., Andersson, E., Jex, I., Silberhorn, C.: Photons walking the line: a quantum walk with adjustable coin operations. Phys. Rev. Lett. 104(5), 050, 502 (2010). doi: 10.1103/PhysRevLett.104.050502
  33. 33.
    Schumacher, B., Werner, R.F.: Reversible quantum cellular automata. arXiv:quant-ph/0405174Google Scholar
  34. 34.
    Segawa E., Konno N.: Limit theorems for quantum walks driven by many coins. Int. J. Quantum Inf. 6, 1231–1243 (2008)zbMATHCrossRefGoogle Scholar
  35. 35.
    Shapira, D., Biham, O., Bracken, A., Hackett, M.: One dimensional quantum walk with unitary noise. Phys. Rev. A 68, 062, 315 (2003)Google Scholar
  36. 36.
    Stein E., Weiss G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, NJ (1971)zbMATHGoogle Scholar
  37. 37.
    Wójcik, A., Kurzyński, T.L.P., Grudka, A., Bednarska, M.: Quasiperiodic dynamics of a quantum walk on the line. Phys. Rev. Lett. 93(18), 180, 601 (2004)Google Scholar
  38. 38.
    Zähringer, F., Kirchmair, G., Gerritsma, R., Solano, E., Blatt, R., Roos, C.F.: Realization of a quantum walk with one and two trapped ions. Phys. Rev. Lett. 104(10), 100, 503 (2010). doi: 10.1103/PhysRevLett.104.100503

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Andre Ahlbrecht
    • 1
  • Christopher Cedzich
    • 1
  • Robert Matjeschk
    • 1
  • Volkher B. Scholz
    • 1
    • 2
  • Albert H. Werner
    • 1
  • Reinhard F. Werner
    • 1
  1. 1.Inst. f. Theoretical PhysicsLeibniz Universität HannoverHannoverGermany
  2. 2.Inst. f. Theoretical PhysicsETH ZurichZurichSwitzerland

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