Homological error correction: Classical and quantum codes

Journal of Mathematical Physics

Volumn 48, Issue 5, 2007, Pages

  Bombin, H ^a   Martin Delgado, M A ^a  

^a UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34249872422     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2731356     Document Type: Article

References (73)

1. R. Landauer, Philos. Trans. R. Soc. London, Ser. A 353, 367 (1995).
2. R. Landauer, Phys. Lett. A 217, 188 (1996).
3. R. Landauer, in Proceedings of Drexel-4 Symposium on Quantum Nonintegrability-Quantum-Classical Correspondence, Philadelphia, PA, 8 September 1994, edited by, D. H. Feng, and, B.-L. Hu, (International Press, Boston, MA, 1997).
4. W. G. Unruh, Phys. Rev. A 51, 992 (1995).
5. P. Shor, Phys. Rev. A 52, 2493 (1995).
6. A. M. Steane, Phys. Rev. Lett. 77, 793 (1996).
7. A. R. Calderbank and P. W. Shor, Phys. Rev. A 54, 1098 (1996).
8. A. M. Steane, Proc. R. Soc. London, Ser. A 452, 2551 (1996).
9. P. W. Shor and J. Preskill, Phys. Rev. Lett. 85, 441 (2000).
10. D. Gottesman, Phys. Rev. A 54, 1862 (1996).
11. A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Phys. Rev. Lett. 78, 405 (1997).
12. C. Bennett, D. DiVincenzo, J. Smolin, and W. Wootters, Phys. Rev. A 54, 3824 (1996).
13. R. Laflamme, C. Miquel, J. P. Paz, and W. Zurek, Phys. Rev. Lett. 77, 198 (1996).
14. E. Knill, e-print arXiv:quant-ph/9608049.
15. A. Klappenecker and M. Roetteler, e-print arXiv:quant-ph/0010082.
16. E. M. Rains, R. H. Hardin, P. W. Shor, and N. J. A. Sloane, Phys. Rev. Lett. 79, 953 (1997).
17. M. B. Ruskai, Phys. Rev. Lett. 85, 194 (2000).
18. H. Pollatsek and M. B. Ruskai, Linear Algebr. Appl. 392, 255 (2004).
19. A. Yu. Kitaev, Ann. Phys. 303, 2 (2003); e-print arXiv:quant-ph/9707021.
20. A. Yu. Kitaev, Ann. Phys. 303, 2 (2003); e-print arXiv:quant-ph/9707021.
21. M. H. Freedman, Proc. Natl. Acad. Sci. U.S.A. 95, 98 (1998).
22. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, J. Math. Phys. 43, 4452 (2002).
23. S. B. Bravyi and A. Yu. Kitaev, e-print arXiv:quant-ph/9811052.
24. M. Levin and X.-G. Wen, Phys. Rev. B 71, 045110 (2005).
25. M. Freedman, C. Nayak, and K. Shtengel, Phys. Rev. Lett. 94, 147205 (2005).
26. P. Fendley and E. Fradkin, Phys. Rev. B 72, 024412 (2005).
27. A. Kitaev, Ann. Phys. 321, 2-111 (2003).
28. M. Freedman, C. Nayak, and K. Shtengel, Phys. Rev. Lett. 94, 066401 (2005).
29. S. H. Simon, N. E. Bonesteel, M. H. Freedman, N. Petrovic, and L. Hormozi, Phys. Rev. Lett. 96, 070503 (2006).
30. X.-G. Wen and Q. Niu, Phys. Rev. B 41, 9377 (1990).
31. M. Freedman, C. Nayak, and K. Walker, Phys. Rev. B 73, 245307 (2006).
32. S. Das Sarma, M. Freedman, and C. Nayak, Phys. Rev. Lett. 94, 166802 (2005).
33. A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006).
34. M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006).
35. F. Verstraete, M. A. Martin-Delgado, and J. I. Cirac, Phys. Rev. Lett. 92, 087201 (2004).
36. J. J. Garcia-Ripoll, M. A. Martin-Delgado, and J. I. Cirac, Phys. Rev. Lett. 93, 250405 (2004).
37. L. M. Duan and G. C. Guo, Phys. Rev. Lett. 79, 1953 (1997).
38. P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997).
39. D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. Lett. 81, 2594 (1998).
40. P. Zanardi and S. Lloyd, Phys. Rev. Lett. 90, 067902 (2003).
41. D. Kribs, R. Laflamme, and D. Poulin, Phys. Rev. Lett. 94, 180501 (2005).
42. L.-M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003); e-print arXiv:cond-mat/0210564.
43. L.-M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003); e-print arXiv:cond-mat/0210564.
44. A. Micheli, G. K. Brennen, and P. Zoller, Nat. Phys. 2, 341-347 (2007).
45. J. K. Pachos, e-print arXiv:quant-ph/0511273.
46. C. Wang, J. Harrington, and J. Preskill, Ann. Phys. (N.Y.) 303, 31 (2003).
47. K. Takeda and H. Nishimori, Nucl. Phys. B 686, 377 (2004).
48. R. W. Ogburn and J. Preskill, Lect. Notes Comput. Sci. 1509, 341 (1999).
49. M. H. Freedman, A. Kitaev, and Z. Wang, Commun. Math. Phys. 227, 587 (2002).
50. M. Freedman, M. Larsen, and Z. Wang, Commun. Math. Phys. 227, 605 (2002).
51. M. H. Freedman, A. Kitaev, M. J. Larsen, and Z. Wang, Bull. Am. Math. Soc. 40, 31 (2003); e-print arXiv:quant-ph/0101025.
52. M. H. Freedman, A. Kitaev, M. J. Larsen, and Z. Wang, Bull. Am. Math. Soc. 40, 31 (2003); e-print arXiv:quant-ph/0101025.
53. P. W. Shor, IEEE Symposium on Foundations of Computer Science, 1996 (unpublished).
54. E. Knill, R. Laflamme, and W. Zurek, e-print arXiv:quant-ph/9610011.
55. D. Gottesman, Phys. Rev. A 57, 127 (1998).
56. D. Aharonov and M. Ben-Or, in Proceedings of the 29th Annual ACM Symposium on Theory of Computing, El Paso, TX, May 1997 (unpublished), pp. 176-188.
57. C. Zalka, e-print arXiv:quant-ph/9612028.
58. J. Preskill, Proc. R. Soc. London, Ser. A 454, 385 (1998).
59. P. Aliferis, D. Gottesman, and J. Preskill, Quantum Inf. Comput. 6, 97 (2006).
60. H. Bombin and M. A. Martin-Delgado, Phys. Rev. A 73, 062303 (2006).
61. M. H. Freedman and D. A. Meyer, e-print arXiv:quant-ph/9810055.
62. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (North-Holland, New York, 1977).
63. D. Welsh, Codes and Cryptography (Oxford University Press, Oxford, 1988).
64. C. E. Shannon, Bell Syst. Tech. J. 27, 379 (1948).
65. H. Bombin and M. A. Martin-Delgado, Phys. Rev. A 72, 032313 (2005).
66. A. Steane, in Introduction to Quantum Computation and Information, edited by, H.-K. Lo, T. Spiller, and, S. Popescu, (World Scientific, Singapore, 1998).
67. J. Preskill, Lecture Notes on Quantum Computation; http://www.theory. caltech.edu/people/preskill/ph229/notes/chap7.ps
68. J. Preskill, Lecture Notes on Quantum Computation; http://www.theory. caltech.edu/people/preskill/ph229/notes/chap7.ps
69. E. Knill and R. Laflamme, Phys. Rev. A 55, 900 (1997).
70. A. Ekert and C. Macchiavello, Phys. Rev. Lett. 77, 2585 (1996).
71. P. J. Giblin, Graphs, Surfaces, and Homology: An Introduction to Algebraic Topology (Chapman and Hall, London, 1981).
72. J. L. Gross and T. W. Tucker, Topological Graph Theory (Wiley, New York, 1987).
73. M. Keyl and R. F. Werner, Lect. Notes Phys. 611, 263 (2002) and references therein.