N-dimensional alternate coined quantum walks from a dispersion-relation perspective

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 87, Issue 2, 2013, Pages

  Roldán, Eugenio ^a   Di Franco, Carlo ^b,c   Silva, Fernando ^a   De Valcárcel, Germán J ^a  

^a UNIVERSITY OF VALENCIA   (Spain)

^b UNIVERSITY OF THE BASQUE COUNTRY UPV EHU   (Spain)

^c QUEEN'S UNIVERSITY BELFAST   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

DISPERSION RELATIONS; IMPLEMENTABILITY; MULTIPARTITE ENTANGLEMENTS; PHYSICAL PHENOMENA; QUANTUM DIFFUSION; QUANTUM WALK; STANDARD DEFINITIONS;

MATHEMATICAL MODELS; PHYSICS;

DISPERSIONS;

EID: 84874526976     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.87.022336     Document Type: Article

References (60)

1. Y. Aharonov, L. Davidovich, and N. Zagury, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.48.1687 48, 1687 (1993);
2. D. Meyer, J. Stat. Phys. JSTPBS 0022-4715 10.1007/BF02199356 85, 551 (1996);
3. J. Watrous, Proceedings of the 33rd Annual ACM Symposium on the Theory of Computing, Heraklion, Crete, Greece, July 06-08, 2001 (ACM Press, New York, 2001), p. 60.
4. E. Farhi and S. Gutmann, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.58.915 58, 915 (1998);
5. A. M. Childs and J. Goldstone, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.70.042312 70, 042312 (2004).
6. F. W. Strauch, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.73.054302 73, 054302 (2006). (Pubitemid 43788059)
7. A. Ambainis, Int. J. Quant. Inf. 0219-7499 10.1142/S0219749903000383 1, 507 (2003);
8. J. Kempe, Contemp. Phys. CTPHAF 0010-7514 10.1080/00107151031000110776 44, 307 (2003); (Pubitemid 36937887)
9. V. Kendon, Math. Struct. Comp. Sci. 17, 1169 (2006);
10. V. Kendon, Philos. Trans. R. Soc., A PTRMAD 1364-503X 10.1098/rsta.2006.1901 364, 3407 (2006); (Pubitemid 44954199)
11. N. Konno, in Quantum Walks, edited by, U. Franz, and, M. Schürmann, Quantum Potential Theory, Lecture Notes in Mathematics Vol. 1954 (Springer, Berlin, 2008).
12. D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, and J. P. Woerdman, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.61.013410 61, 013410 (1999).
13. B. Do, J. Opt. Soc. Am. B JOBPDE 0740-3224 10.1364/JOSAB.22.000499 22, 499 (2005); (Pubitemid 40377755)
14. C. A. Ryan, M. Laforest, J. C. Boileau, and R. Laflamme, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.72.062317 72, 062317 (2005); (Pubitemid 43272723)
15. P. Zhang, X. F. Ren, X. B. Zou, B. H. Liu, Y. F. Huang, and G. C. Guo, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.75.052310 75, 052310 (2007); (Pubitemid 47014624)
16. P. H. Souto Ribeiro, S. P. Walborn, C. Raitz, L. Davidovich, and N. Zagury, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.78.012326 78, 012326 (2008);
17. M. Karski, Science SCIEAS 0036-8075 10.1126/science.1174436 325, 174 (2009);
18. H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.100.170506 100, 170506 (2008); (Pubitemid 351624659)
19. H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett. 103.090504 103, 090504 (2009);
20. F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.104.100503 104, 100503 (2010);
21. M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik, and A. G. White, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.104.153602 104, 153602 (2010).
22. A. Schreiber, K. N. Cassemiro, V. Potocek, A. Gabris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.104.050502 104, 050502 (2010);
23. A. Schreiber, K. N. Cassemiro, V. Potocek, A. Gabris, I. Jex, and C. Silberhorn, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.106.180403 106, 180403 (2011).
24. A. Peruzzo, Science SCIEAS 0036-8075 10.1126/science.1193515 329, 1500 (2010);
25. J. C. F. Matthews, arXiv: 1106.1166 [quant-ph];
26. J. O. Owens, New J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/13/7/075003 13, 075003 (2011);
27. L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.108. 010502 108, 010502 (2012).
28. T. D. Mackay, S. D. Bartlett, L. T. Stephenson, and B. C. Sanders, J. Phys. A: Math. Gen. 1361-6447 10.1088/0305-4470/35/12/304 35, 2745 (2002). (Pubitemid 37023952)
29. C. Moore and A. Russell, in Proceedings of RANDOM, 2002, edited by, J. D. P. Rolim, and, P. Vadham, (Springer, Cambridge, MA, 2002), pp. 164-178.
30. E. Roldán and J. C. Soriano, J. Mod. Opt. JMOPEW 0950-0340 10.1080/09500340500309873 52, 2649 (2005). (Pubitemid 43024360)
31. A. Schreiber, Science SCIEAS 0036-8075 10.1126/science.1218448 336, 55 (2012).
32. B. Tregenna, New. J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/5/1/383 5, 83 (2003);
33. I. Carneiro, New. J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/7/1/156 7, 156 (2005); (Pubitemid 41102095)
34. A. C. Oliveira, R. Portugal, and R. Donangelo, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.74.012312 74, 012312 (2006); (Pubitemid 44079174)
35. K. Watabe, N. Kobayashi, M. Katori, and N. Konno, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.77.062331 77, 062331 (2008);
36. M. Stefanák, Phys. Scr., T PHSTBO 0031-8949 10.1088/0031-8949/ 2010/T140/014035 140, 014035 (2010);
37. N. Shenvi, J. Kempe, and K. BirgittaWhaley, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.67.052307 67, 052307 (2003).
38. N. Inui, Y. Konishi, and N. Konno, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.69.052323 69, 052323 (2004).
39. C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.106.080502 106, 080502 (2011).
40. C. Di Franco, M. Mc Gettrick, T. Machida, and Th. Busch, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.84.042337 84, 042337 (2011).
41. M. V. Berry and M. Wilkinson, Proc. R. Soc. A 1364-5021 10.1098/rspa.1984.0022 392, 15 (1984).
42. M. V. Berry and M. R. Jeffrey, Prog. Opt. 9780-4445 10.1016/S0079- 6638(07)50002-8 50, 13 (2007).
43. G. J. de Valcárcel, E. Roldán, and A. Romanelli, New J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/12/12/123022 12, 123022 (2010).
44. P. L. Knight, E. Roldán, and J. E. Sipe, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.68.020301 68, 020301 (2003);
45. P. L. Knight, E. Roldán, and J. E. Sipe, J. Mod. Opt. 0950-0340 10.1080/09500340408232489 51, 1761 (2006).
46. G. J. de Valcárcel, M. Hinarejos, E. Roldán, A. Pérez, and A. Romanelli, arXiv: 1212.3600 [quant-ph].
47. C. A. Mead and D. G. Truhlar, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.437734 70, 2284 (1979);
48. L. S. Cederbaum, R. S. Friedman, V. M. Ryaboy, and N. Moiseyev, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.90.013001 90, 013001 (2003).
49. A. Bostwick, New J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/9/10/385 9, 385 (2007). (Pubitemid 350046212)
50. A. H. Castro Neto, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys. 81.109 81, 109 (2009).
51. M. Zhang, H. H. Hung, Ch. Zhang, and C. Wu, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.83.023615 83, 023615 (2011).
52. S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.108.140405 108, 140405 (2012).
53. Y. Omar, N. Paunković, L. Sheridan, and S. Bose, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.74.042304 74, 042304 (2006);
54. S. E. Venegas-Andraca and S. Bose, arXiv: 0901.3946 [quant-ph];
55. P. Xue and B. C. Sanders, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.85.022307 85, 022307 (2012).
56. R. Horodecki, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.81.865 81, 865 (2009).
57. C. Sabin and G. Garcia-Alcaine, Eur. Phys. J. D EPJDF6 1434-6060 10.1140/epjd/e2008-00112-5 48, 435 (2008).
58. S. Lee, D. P. Chi, S. D. Oh, and J. Kim, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.68.062304 68, 062304 (2003).
59. P. L. Knight, J. E. Sipe, and E. Roldán, Opt. Commun. OPCOB8 0030-4018 10.1016/j.optcom.2003.09.024 227, 147 (2003).
60. The multiplexing requires that the frequency displacements corresponding to different dimensions have to be very different. (In fact, this difference determines the maximum number of steps that can be implemented; see the discussion in Ref.). Moreover, the frequency shifts should be done with a high repetition rate (larger than the inverse of the cavity round-trip time). All this could be very demanding for actual EOMs.