Random Time-Dependent Quantum Walks

Communications in Mathematical Physics

Volumn 307, Issue 1, 2011, Pages 65-100

  Joye, Alain ^a  

^a UNIV GRENOBLE ALPES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 80052478610     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-011-1297-7     Document Type: Article

References (37)

1. Aharonov Y., Davidovich L., Zagury N.: Quantum random walks. Phys. Rev. A 48, 1687-1690 (1993).
2. Ahlbrecht A., Vogts H., Werner A. H., Werner R. F.: Asymptotic evolution of quantum walks with random coin. J. Math. Phys. 52, 042201 (2011).
3. Ambainis, A., Aharonov, D., Kempe, J., Vazirani, U.: Quantum Walks on Graphs. In: Proc. 33rd ACM STOC, 2001, pp. 50-59.
4. Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of SODA'05, 2005 pp. 1099-1108.
5. Asch J., Bourget O., Joye A.: Localization Properties of the Chalker-Coddington Model. Ann. H. Poincaré 11(7), 1341-1373 (2010).
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. John Wiley and Sons, New York (1968).
8. Bourget O., Howland J. S., Joye A.: Spectral analysis of unitary band matrices. Commun. Math. Phys 234, 191-227 (2003).
9. Blatter G., Browne D.: Zener tunneling and localization in small conducting rings. Phys. Rev. B 37, 3856 (1988).
10. Bruneau L., Joye A., Merkli M.: Infinite Products of Random Matrices and Repeated Interaction Dynamics. Ann. Inst. Henri Poincaré (B) Prob. Stat. 46, 442-464 (2010).
11. Chalker J. T., Coddington P. D.: Percolation, quantum tunneling and the integer Hall effect. J. Phys. C 21, 2665-2679 (1988).
12. de Oliveira C. R., Simsen M. S.: A Floquet Operator with Purely Point Spectrum and Energy Instability. Ann. H. Poincaré 7, 1255-1277 (2008).
13. Dembo A., Zeitouni O.: Large Deviations Techniques and Applications. Springer, Berlin-Heidelberg-New york (1998).
14. Hamza E., Joye A., Stolz G.: Dynamical Localization for Unitary Anderson Models. Math. Phys., Anal. Geom. 12, 381-444 (2009).
15. Hamza E., Kang Y., Schenker J.: Diffusive propagation of wave packets in a fluctuating periodic potential. Lett. Math. Phys 95, 53-66 (2011).
16. Joye A., Merkli M.: Dynamical Localization of Quantum Walks in Random Environments. J. Stat. Phys. 140, 1025-1053 (2010).
17. Kang Y., Schenker J.: Diffusion of wave packets in a Markov random potential. J. Stat. Phys. 134, 1005-1022 (2009).
18. Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin-Heidelberg-New York (1980).
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.: Quantum Walks. In: Quantum Potential Theory. Franz, Schürmann eds., Lecture Notes in Mathematics, 1954, Berlin-Heidelberg-New York: Springer, 2009, pp. 309-452.
24. Kosk J., Buzek V., Hillery M.: Quantum walks with random phase shifts. Phys. Rev. A 74, 022310 (2006).
25. Landim, C.: Central Limit Theorem for Markov Processes. In: From Classical to Modern Probability CIMPA Summer School 2001, Picco, Pierre; San Martin, Jaime (Eds.), Progress in Probability 54, Basel: Birkhaüser, 2003, pp. 147-207.
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. 20th SODA Philadelphia PA: SIAM, 2009, pp. 86-95.
28. Meyer D.: From quantum cellular automata to quantum lattice gases. J. Stat. Phys. 85, 551-574 (1996).
29. Pillet C. A.: Some Results on the Quantum Dynamics of a Particle in a Markovian Potential. Commun. Math. Phys. 102, 237-254 (1985).
30. Ryu J.-W., Hur G., Kim S. W.: Quantum Localization in Open Chaotic Systems. Phys. Rev. E 78, 037201 (2008).
31. Santha, M.: Quantum walk based search algorithms, 5th TAMC (Xian, 2008), LNCS 4978, Berlin-Heidelberg-New York: Springer Verlag, 2008, pp. 31-46.
32. Shapira D., Biham O., Bracken A. J., Hackett M.: One dimensional quantum walk with unitary noise. Phys. Rev. A 68, 062315 (2003).
33. Shikano Y., Katsura H.: Localization and fractality in inhomogeneous quantum walks with self-duality. Phys. Rev. E 82, 031122 (2010).
34. Shenvi N., Kempe J., Whaley K. B.: Quantum random-walk search algorithm. Phys. Rev. A 67, 052307 (2003).
35. Yin Y., Katsanos D. E., Evangelou S. N.: Quantum Walks on a Random Environment. Phys. Rev. A 77, 022302 (2008).
36. Zhan X.: Matrix Inequalities. LNM 1790. Springer, Berlin-Heidelberg-New York (2002).
37. 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, 100503 (2010).