Limit distributions of two-dimensional quantum walks

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 77, Issue 6, 2008, Pages

  Watabe, Kyohei ^a   Kobayashi, Naoki ^a   Katori, Makoto ^a   Konno, Norio ^b  

^a CHUO UNIVERSITY   (Japan)

^b YOKOHAMA NATIONAL UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL METHODS; COMPUTER NETWORKS; COMPUTER SIMULATION; MATHEMATICAL MODELS; TWO DIMENSIONAL;

(001) PARAMETER; A PLANE; AMERICAN PHYSICAL SOCIETY (APS); CONVERGENCE (MATHEMATICS); DENSITY FUNCTIONS; DISCRETE TIME (DT); JOINT MOMENTS; LIMIT DISTRIBUTIONS; NUMERICA L RESULTS; PROBABILITY DENSITIES; QUANTUM WALKS; SQUARE LATTICES; TIME LIMIT; TWO-DIMENSIONAL (2D);

PROBABILITY DISTRIBUTIONS;

EID: 45749116574     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.77.062331     Document Type: Article

References (37)

1. B. C. Travaglione and G. J. Milburn, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.65.032310 65, 032310 (2002).
2. J. Kempe, Contemp. Phys. CTPHAF 0010-7514 10.1080/00107151031000110776 44, 307 (2003).
3. A. Ambainis, Int. J. Quantum Inf. ZZZZZZ 0219-7499 1, 507 (2003).
4. T. A. Brun, H. A. Carteret, and A. Ambainis, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.67.052317 67, 052317 (2003).
5. V. M. Kendon, Int. J. Quantum Inf. ZZZZZZ 0219-7499 4, 791 (2006).
6. Y. Aharonov, L. Davidovich, and N. Zagury, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.48.1687 48, 1687 (1993).
7. D. A. Meyer, J. Stat. Phys. JSTPBS 0022-4715 10.1007/BF02199356 85, 551 (1996).
8. A. Nayak and A. Vishwanath, e-print arXiv:quant-ph/0010117.
9. A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (ACM Press, New York, 2001), pp. 37-49.
10. N. Konno, Quantum Walks, Lecture at the School "Quantum Potential Theory: Structure and Applications to Physics" held at the Alfried Krupp Wissenschaftskolleg, Greifswald, 26 February-9 March 2007, No. 2 (Reihe Mathematik, Ernst-Moritz-Arndt-Universität Greifswald, Greifswald, 2007), the lecture note is available at http://www.math-inf.uni-greifswald.de/algebra/ qpt/konno-26nov2007, and will be presented in Springer Lecture Notes in Mathematics.
11. Precisely speaking, the theory of quantum walks has been divided into the discrete-time version and the continuous-time version. In the present paper we focus on the discrete-time models. The study on the connection of these two versions is itself interesting and important. See, for example, F. W. Strauch, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.74.030301 74, 030301 (R) (2006).
12. N. Konno, Quantum Inf. Process. QIPUAT 1570-0755 10.1023/A:1023413713008 1, 345 (2002).
13. N. Konno, J. Math. Soc. Jpn. NISUBC 0025-5645 10.2969/jmsj/1150287309 57, 1179 (2005).
14. G. Grimmett, S. Janson, and P. F. Scudo, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.69.026119 69, 026119 (2004).
15. M. Katori, S. Fujino, and N. Konno, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.72.012316 72, 012316 (2005).
16. T. Miyazaki, M. Katori, and N. Konno, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.76.012332 76, 012332 (2007).
17. M. Sato, N. Kobayashi, M. Katori, and N. Konno, e-print arXiv:quant-ph/0802.
18. T. D. Mackay, S. D. Bartlett, L. T. Stephenson, and B. C. Sanders, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/35/12/304 35, 2745 (2002).
19. B. Tregenna, W. Flanagan, R. Maile, and V. Kendon, New J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/5/1/383 5, 83 (2003).
20. I. Carneiro, M. Loo, X. Xu, M. Girerd, V. Kendon, and P. L. Knight, New J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/7/1/156 7, 156 (2005).
21. S. E. Venegas-Andraca, J. L. Ball, K. Burnett, and S. Bose, New J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/7/1/221 7, 221 (2005).
22. O. Mülken, A. Volta, and A. Blumen, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.72.042334 72, 042334 (2005).
23. L. K. Grover, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.79. 325 79, 325 (1997).
24. N. Shenvi, J. Kempe, and K. Birgitta Whaley, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.67.052307 67, 052307 (2003).
25. A. M. Childs and J. Goldstone, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.70.022314 70, 022314 (2004).
26. A. M. Childs and J. Goldstone, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.70.042312 70, 042312 (2004).
27. A. Tulsi, e-print arXiv:quant-ph/0801.0497.
28. N. Inui, Y. Konishi, and N. Konno, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.69.052323 69, 052323 (2004).
29. Localization of quantum walk studied by Inui and co-workers and in the present paper is not directly related to the Anderson localization. If we consider quantum walks with disorder, however, the process is closely related to Anderson's model. In the continuous-time quantum-walk version, the Anderson localization was discussed in O. Mülken, V. Bierbaum, and A. Blumen, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.75.031121 75, 031121 (2007).
30. A. C. Oliveira, R. Portugal, and R. Donangelo, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.74.012312 74, 012312 (2006).
31. H. Georgi, Lie Algebras in Particle Physics, 2nd ed. (Perseus Books, Reading, 1999).
32. E. Roldán and J. C. Soriano, J. Mod. Opt. JMOPEW 0950-0340 10.1080/09500340500309873 52, 2649 (2005).
33. K. Eckert, J. Mompart, G. Birkl, and M. Lewenstein, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.72.012327 72, 012327 (2005).
34. S. Fujiwara, H. Osaki, I. M. Buluta, and S. Hasegawa, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.72.032329 72, 032329 (2005).
35. R. Côté, A. Russell, E. E. Eyler, and P. L. Gould, New J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/8/8/156 8, 156 (2006).
36. O. Mülken, A. Blumen, T. Amthor, C. Giese, M. Reetz-Lamour, and M. Weidemüller, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.99. 090601 99, 090601 (2007).
37. M. J. Ablowitz and A. S. Fokas, Complex Variables, Introduction and Applications, 2nd ed. (Cambridge University Press, Cambridge, 2003).