Enseñanza

 

Introduction to error correcting codes in quantum computers

 

P. J. Salas-Peralta

 

Departamento de Tecnologías Especiales Aplicadas a la Telecomunicación, Universidad Politécnica de Madrid, Ciudad Universitaria s/n, 28040 Madrid, e-mail: psalas@etsit.upm.es

 

Recibido el 25 de octubre de 2005;
aceptado el 7 de marzo de 2006

 

Abstract

The goal of this paper is to review the theoretical basis for achieving a faithful quantum information transmission and processing in the presence of noise. Initially, encoding and decoding, implementing gates and quantum error correction will be considered error-free. Finally, we shall relax this non-realistic assumption, introducing the quantum fault-tolerant concept. The existence of an error threshold permits us to conclude that there is no physical law preventing a quantum computer from being built. An error model based on the depolarising channel will be able to provide a simple estimate of the storage or memory computation error threshold: η_th < 5.2 10^-5. The encoding is made by means of the [[7,1,3]] Calderbank-Shor-Steane quantum code, and Shor's fault-tolerant method is used to measure the stabiliser's generators.

Keywords: Quantum error correcting codes; decoherence; quantum computation.

 

Resumen

El objetivo de este artículo es la revisión de los fundamentos teóricos que permiten una correcta transmisión y procesado de la información cuántica en presencia de ruido. Inicialmente, los procesos de codificación, decodificación, aplicación de puertas y corrección de errores se consideraran sin error. Finalmente relajaremos esta consideración no realista, lo que conducirá al concepto de tolerancia a fallos. La existencia de un umbral de error permite concluir que no hay ninguna ley física que impida construir un ordenador cuántico. Mediante un modelo de error basado en un canal despolarizante, se hará una estimación simple para el umbral de los errores de memoria: η_th < 5.2 10^-5. La codificación se realiza mediante un código cuántico [[7,1,3]] de Calderbank-Shor-Steane, y se usa el método de Shor tolerante a fallos para medir los generadores del estabilizador.

Descriptores: Códigos correctores de errores cuánticos; decoherencia; computación cuántica.

 

PACS: 0367-a; 0367Lx

 

DESCARGAR ARTÍCULO EN FORMATO PDF

 

References

1. A. Galindo and M.A. Martín-Delgado, Rev. Mod. Phys. 74. 347 (2002).         [ Links ]

2. W.G. Unruh, Phys. Rev. A 51 (1995) 992.         [ Links ]

3. Some general references for the whole paper are the following: A.M. Steane, Rep. Prog. Phys. 61 (1998) 117;         [ Links ] E. Knill and R. Laflamme, Phys. Rev. A 55 (1997) 900 (quant-ph/9604034);         [ Links ] E. Knill et al., Technical Report LAUR-01-6115, Los Alamos National Laboratory (2002) (quant-ph/0207170);         [ Links ] M.A. Nielsen and I.L. Chuang. Quantum computation and quantum information, 2nd edn. (Cambridge University Press, Cambridge, 2001);         [ Links ] J. Gruska. Quantum computing, 1st ed. (McGraw-Hill, UK, 1999);         [ Links ] J. Stolze and D. Suter. Quantum Computing (Wiley-VCH, Weinheim, 2004);         [ Links ] G.P. Berman, G.D. Doolen, R. Mainieri, and V.I. Tsifrinovich. Introduction to quantum computers, 1st ed. (World Scientific, Singapore, 1998);         [ Links ] A.O. Pittenger. An introduction to quantum computing algorithms, 1st ed. (Birkhauser, Boston, 1999).         [ Links ] D. Gottesman, in Quantum computation: A grand mathematical challenge for the twenty-first century, edited by S. Lomonaco, 1st ed. (American Mathematical Society, Washington DC, 2000);         [ Links ] J. Preskill. Lecture notes, http://www.theory.caltech.edu/people/preskill/ph229/ (1998).         [ Links ] H. Chang. A brief survey of quantum error-correcting codes, Technical Report, CMU-QTG-TR-2005-01, Carnegie Mellon University, May 2005.         [ Links ]

4. S. Kak, Found. of Phys. 29 (1999) 267.         [ Links ]

5. W.K. Wootters and W.H. Zurek, Nature 299 (1982) 802.         [ Links ]

6. A. Ekert and Ch. Macchiavello, Acta Phys. Polon. A 93 (1998) 63 (quant-ph/9904070).         [ Links ]

7. E. Knill, R. Laflamme and W.H. Zurek, Science 279 (1998) 342;         [ Links ] E. Knill, R. Laflamme, and W.H. Zurek, Proc. R. Soc. of London A 454 (1998) 365 (quant-ph/9702058).         [ Links ]

8. L.K. Grover. Proceedings of 28th Annual ACM Symposium on Theory of Computation (Philadelphia USA, 1996) p. 212;         [ Links ] L.K. Grover, Phys. Rev. Lett. 79 (1997) 325;         [ Links ] L.K. Grover, Am. J. Phys. 69 (2001) 769.         [ Links ]

9. A. Barenco et al., Phys. Rev. A 52 (1995) 3457.         [ Links ]

10. C. Lavor, L.R.U. Manssur, and R. Portugal. quant-ph/0301079.         [ Links ]

11 . L. Vaidman, L. Goldenberg, and S. Wisner, Phys. Rev. A 54 (1996) R1745;         [ Links ] L. Duan and G. Guo, Phys. Rev. A 58 (1998) 3491 (quant-ph/9804014).         [ Links ]

12. W. Zurek, Phys. Rev. Lett. 53 (1984) 391.         [ Links ]

13. A. Barenco et al., SIAM J. of Comp. 26 (1997) 1541 (quant-ph/9604028);         [ Links ] A. Peres, Int. J. Theor. Phys. 38 (1999) 799 (quant-ph/9702001).         [ Links ]

14. G.M. Palma, K.A. Suominen, and A.K. Ekert, Proc. Roy. Soc. London A 452 (1996) 567 (quant-ph/9702001);         [ Links ] L. Duan and G. Duo, Phys. Lett. A 255 (1999) 209 (quant-ph/9809057).         [ Links ]

15. S. De Filippo, Phys. Rev. A 62 (2000) 52307;         [ Links ] P. Zanardi and M. Rasetti, Mod. Phys. Lett. B 11(1997) 1085;         [ Links ] D.A. Lidar, I.L. Chuang, and K.B. Whaley, Phys. Rev. Lett. 81 (1998) 2594;         [ Links ] D.A. Lidar, in Decoherence-free subspaces and subsystems, edited by F. Benatti, R Floreanini (Springer Lecture Notes in Physics, Berlin, 2003).         [ Links ]

16. D. Welsh. Codes and cryptography, 6th ed. (Clarendon Press, Oxford, 1996).         [ Links ]

17. P.W. Shor, Phys. Rev. A 52 (1995) R2493.         [ Links ]

18. A.R. Calderbank and P. Shor, Phys. Rev. A 54 (1996) 1098; A.R. Calderbank, E.M. Rains, P.W. Shor, and N.J.A. Sloane, Phys. Rev. Lett. 78 (1997) 405.         [ Links ]

19. A.M. Steane, Proc. R. Soc. of London A, 452 2551(1996); A.M. Steane, Phys. Rev. Lett. 78 (1997) 2252; A.M. Steane, Phys. Rev. Lett. 77 (1996) 793; A.M. Steane, Phys. Rev. A 54 (1996) 4741; A.M. Steane, Fortsch. Phys. 46 (1998) 443.         [ Links ]

20. F.J. MacWilliams and N.J.A. Sloane, The theory of error-correcting codes, 6th ed. (North-Holland, Amsterdam, 1988).         [ Links ]

21. D. Gottesman. Ph.D. thesis, California Institute of Technology (1997) (quant-ph/9705052).         [ Links ]

22. This code was independently discovered by C.H. Bennett, D. DiVincenzo, J.A. Smolin, and W.K. Wootters, Phys. Rev. A 54 (1996) 3824 (quant-ph/9604024) and by R. Laflamme, C. Miquel, J.P. Paz, and W.H. Zurek, Phys. Rev. Lett. 77 (1996) 198 (quant-ph/9602019).         [ Links ]

23. D. Gottesman, Phys. Rev. A 54 (1996) 1862.         [ Links ]

24. A.R. Calderbank, E.M. Rains, P.W. Shor, and N.J.A. Sloane, IEEE Trans. Inform. Theory 44 (1998) 1369;         [ Links ] G. Hoehn. Mathematische Annalen 327 (2003) 227. Technical report mat.CO/0005266.         [ Links ]

25. A. Ketkar, A. Klappenecker, S. Kumar, and P.K. Sarvepalli. Technical Report quant-ph/0508070.         [ Links ]

26. A.M. Steane in Introduction to quantum computation and information, edited by H. Lo, S. Popescu, and T. Spiller, 3rd edn. (World Scientific, Singapore, 2000).         [ Links ]

27. D.W. Kribs, Technical report math.OA/0506491 (to appear in Proceedings of the 25th Meeting of the Great Plains Operator Theory Symposium);         [ Links ] D.W. Kribs, R. Laflamme, D. Poulin, and M. Lesosky, Technical report quant-ph/0504189 (to appear in Quantum Information & Computation);         [ Links ] D.W. Kribs, Linear Algebra and its Applications 400 (2005) 147.         [ Links ]

28. J. Preskill, Physics Today 52 (1999) 24;         [ Links ] J. Preskill, in Introduction to quantum computation and information, edited by H. Lo, S. Popescu, and T. Spiller, 3rd ed. (World Scientific, Singapore, 2000);         [ Links ] D. Aharonov. Proceedings of 29th Annual ACM Symposium on Theory of Computing (STOC), El Paso, Texas, (1997) (quant-ph/9611025);         [ Links ] D. Gottesman, J. of Mod. Opt. 47 (2000) 333 (quant-ph/9903099);         [ Links ] A. Steane, Nature 399 (1999) 124;         [ Links ] D.P. DiVincenzo and P.W. Shor, Phys. Rev. Lett. 77 (1996) 3260;         [ Links ] D. Gottesman, Phys. Rev. A 57 (1998) 127.         [ Links ] D. Gottes-man, Technical Report quant-ph/0507174.         [ Links ]

29. W.H. Zurek and R. Laflamme, Phys. Rev. Lett. 77 (1996) 4683.         [ Links ]

30. P.J. Salas and A.L. Sanz, Phys. Rev. A 66 (2002) 022302 (quant-ph/0207068) P.         [ Links ]J. Salas and A.L. Sanz, Phys. Rev. A 69 (2004) 052322 (quant-ph/0405012).         [ Links ]

31. P. Shor, 37th symposium on Found. of Comp. Science (Los Alamitos, California, 1996) p. 56, (quant-ph/9605011).         [ Links ]

32. E. Knill and R. Laflamme, Technical Report LAUR-96-2808, Los Alamos National Laboratory (1996) (quant-ph/9608012).         [ Links ]

33. E. Knill, R. Laflamme, and W. Zurek, Technical Report LAUR-96-2199, Los Alamos National Laboratory (1996) (quant-ph/9608012);         [ Links ] C. Zalka. quant-ph/9612028 (1996).         [ Links ]

34. B.W. Reichardt, Technical Report quant-ph/0406025.         [ Links ]