Dynamical Localization of Quantum Walks in Random Environments

Journal of Statistical Physics

Volumn 140, Issue 6, 2010, Pages 1025-1053

  Joye, Alain ^a   Merkli, Marco ^b  

^a CNRS   (France)

^b MEMORIAL UNIVERSITY OF NEWFOUNDLAND   (Canada)

Author keywords

Aizenman Molchanov method; Dynamical Anderson localization; Fractional moment estimates; Lyapunov exponent; Quantum walk; Random environment

Indexed keywords

EID: 77956172559     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-010-0047-0     Document Type: Article

References (35)

1. Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687-1690 (1993).
2. Aizenman, M., Molchanov, S.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245-278 (1993).
3. Ambainis, A., Aharonov, D., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of 33rd ACM STOC, pp. 50-59 (2001).
4. Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of SODA'05, p1099-1108 (2005).
5. Ao, P.: Absence of localization in energy space of a Bloch electron driven by a constant electric force. Phys. Rev. B 41, 3998-4001 (1989).
6. Asch, J., Duclos, P., Exner, P.: Stability of driven systems with growing gaps, quantum rings, and Wannier ladders. J. Stat. Phys. 92, 1053-1070 (1998).
7. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968).
8. Bougerol, O., Lacroix, J.: Products of Random Matrices with Applications to Schrödinger Operators. Birkhäuser Progress in Probability and Statistics. Birkhäuser, Basel (1985).
9. Bourget, O., Howland, J. S., Joye, A.: Spectral analysis of unitary band matrices. Commun. Math. Phys. 234, 191-227 (2003).
10. Blatter, G., Browne, D.: Zener tunneling and localization in small conducting rings. Phys. Rev. B 37, 3856 (1988).
11. Cantero, M. J., Grünbaum, F. A., Morales, L., Velàzquez, L.: Matrix valued Szegö polynomials and quantum random walks. Commun. Pure Appl. Math. 63, 464-507 (2009).
12. Carmona, R., Klein, A., Martinelli, F.: Anderson localization for Bernoulli and other singular potentials. Commun. Math. Phys. 108, 41-66 (1987).
13. Chen, A., Renshaw, E.: The Gillis-Domb-Fisher correlated random walk. J. Appl. Probab. 29, 792-813 (1992).
14. de Oliveira, C. R., Simsen, M. S.: A Floquet operator with purely point spectrum and energy instability. Ann. Henri Poincaré 7, 1255-1277 (2008).
15. Hamza, E.: Localization properties for the unitary Anderson model. PhD thesis, University of Alabama at Birmingham (2007). https://www. mhsl. uab. edu/dt/2008r/hamza. pdf.
16. Hamza, E., Stolz, G.: Lyapunov exponents for unitary Anderson models. J. Math. Phys. 48, 043301 (2007).
17. Hamza, E., Joye, A., Stolz, G.: Dynamical localization for unitary Anderson models. Math. Phys. Anal. Geom. 12, 381-444 (2009).
18. Joye, A.: Fractional moment estimates for random unitary operators. Lett. Math. Phys. 72(1), 51-64 (2005).
19. Karski, M., Förster, L., Chioi, J. M., Streffen, A., Alt, W., Meschede, D., Widera, A.: Quantum walk in position space with single optically trapped atoms. Science 325, 174-177 (2009).
20. Keating, J. P., Linden, N., Matthews, J. C. F., Winter, A.: Localization and its consequences for quantum walk algorithms and quantum communication. Phys. Rev. A 76, 012315 (2007).
21. Kempe, J.: Quantum random walks-an introductory overview. Contemp. Phys. 44, 307-327 (2003).
22. Konno, N.: One-dimensional discrete-time quantum walks on random environments. Quantum Inf. Process. 8, 387-399 (2009).
23. Konno, N.: Localization of an inhomogeneous discrete-time quantum walk on the line. Quantum Inf. Process. (to appear).
24. Konno, N.: Quantum walks. In: Schürmann, F. (ed.) Quantum Potential Theory. Lecture Notes in Mathematics, vol. 1954, pp. 309-452. Springer, Berlin (2009).
25. Kos'k, J., Buzek, V., Hillery, M.: Quantum walks with random phase shifts. Phys. Rev. A 74, 022310 (2006).
26. Lenstra, D., van Haeringen, W.: Elastic scattering in a normal-metal loop causing resistive electronic behavior. Phys. Rev. Lett. 57, 1623-1626 (1986).
27. Magniez, F., Nayak, A., Richter, P. C., Santha, M.: On the hitting times of quantum versus random walks. In: 20th SODA, pp. 86-95 (2009).
28. Meyer, D.: From quantum cellular automata to quantum lattice gases. J. Stat. Phys. 85, 551-574 (1996).
29. Renshaw, E., Henderson, R.: The correlated random walk. J. Appl. Probab. 18, 403-414 (1981).
30. Ryu, J.-W., Hur, G., Kim, S. W.: Quantum localization in open chaotic systems. Phys. Rev. E 037201 (2008).
31. Santha, M.: Quantum walk based search algorithms. In: 5th TAMC. LNCS, vol. 4978, pp. 31-46. Springer, Berlin (2008).
32. Shapira, D., Biham, O., Bracken, A. J., Hackett, M.: One dimensional quantum walk with unitary noise. Phys. Rev. A 68, 062315 (2003).
33. Shenvi, N., Kempe, J., Whaley, K. B.: Quantum random-walk search algorithm. Phys. Rev. A 67, 052307 (2003).
34. Yin, Y., Katsanos, D. E., Evangelou, S. N.: Quantum walks on a random environment. Phys. Rev. A 77, 022302 (2008).
35. Zähringer, F., Kirchmair, G., Gerritsma, R., Solano, E., Blatt, R., Roos, C. F.: Realization of a quantum walk with one and two trapped ions. arXiv: 0911. 1876v1 [quant-ph].