Encyclopedia of Complexity and Systems Science

2009 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Quantum Error Correction and Fault Tolerant Quantum Computing

  • Markus Grassl
  • Martin Rötteler
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30440-3_435

Definition of the Subject

Quantum error correction offers a solution to the problem of protecting quantum systems against noise induced by interactions with theenvironment or caused by imperfect control of the system. The need for error correction arises not only in communication, when quantum information is sentover some distance, but also in locally, when storing and processing quantum information. Fault-tolerant quantum computing builds on quantum errorcorrection and denotes techniques that allow computations to be performed on a quantum system with faulty gates as well as storage errors. Withoutmechanisms for quantum error correction and fault-tolerance, quantum computing would be impossible even for moderate error rates.

The idea of quantum error correction was first conceived in a paper [64] by Shor in 1995in which a particular quantum code was given that encodes one quantum bit (qubit) into nine quantum bits, while being able to correct against onearbitrary error on one of...

This is a preview of subscription content, log in to check access.

Bibliography

  1. 1.
    Aaronson S, Gottesman D (2004) Improved simulation of stabilizer circuits. Phys Rev A 70(5):052328ADSGoogle Scholar
  2. 2.
    Aharonov D, Ben-Or M (1997) Fault-tolerant quantum computation with constant error. In: Proceedings of 29th ACM symposium on theory of computing (STOC'97). ACM, El Paso, pp 176–188Google Scholar
  3. 3.
    Aliferis P, Gottesman D, Preskill J (2006) Quantum accuracy threshold for concatenated distance-3 codes. Quantum Inf Comput 6(2):97–165MathSciNetMATHGoogle Scholar
  4. 4.
    Aliferis P, Gottesman D, Preskill J (2008) Accuracy threshold for postselected quantum computation. Quantum Inf Comput 8:181–244MathSciNetMATHGoogle Scholar
  5. 5.
    Anders S, Briegel HJ (2006) Fast simulation of stabilizer circuits using a graph-state representation. Phys Rev A 73:022334ADSGoogle Scholar
  6. 6.
    Ashikhmin A, Knill E (2001) Nonbinary quantum stabilizer codes. IEEE Trans Inf Theory 47(7):3065–3072MathSciNetMATHGoogle Scholar
  7. 7.
    Avizienis A, Laprie JC, Randell B, Landwehr C (2004) Basic concepts and taxonomy of dependable and secure computing. IEEE Trans Dependable Secur Comput 1(1):11–33Google Scholar
  8. 8.
    Bacon D (2006) Operator quantum error-correcting subsystems for self-correcting quantum memories. Phys Rev A 73:012340ADSGoogle Scholar
  9. 9.
    Barenco A, Berthiaume A, Deutsch D, Ekert A, Macciavello C (1997) Stabilization of quantum computations by symmetrization. SIAM J Comput 26(5):1541–1557MathSciNetMATHGoogle Scholar
  10. 10.
    Bennett CH, Brassard G, Crepeau C, Jozsa R, Peres A, Wootters W (1993) Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys Rev Lett 70(13):1895–1899MathSciNetADSMATHGoogle Scholar
  11. 11.
    Bennett CH, DiVincenzo DP, Smolin JA, Wootters WK (1996) Mixed-state entanglement and quantum error correction. Phys Rev A 54(5):3824–3851MathSciNetADSGoogle Scholar
  12. 12.
    Berlekamp ER, McEliece RJ, van Tilborg HCA (1978) On the inherent intractability of certain coding problems. IEEE Trans Inf Theory 24(3):384–386MATHGoogle Scholar
  13. 13.
    Brun T, Devetak I, Hsieh MH (2006) Correcting quantum errors entanglement. Science 314(5798):436–439MathSciNetADSMATHGoogle Scholar
  14. 14.
    Calderbank AR, Rains EM, Shor PW, Sloane NJA (1998) Quantum error correction via codes over GF(4). IEEE Trans Inf Theory 44(4):1369–1387MathSciNetMATHGoogle Scholar
  15. 15.
    Calderbank AR, Shor PW (1996) Good quantum error-correcting codes exist. Phys Rev A 54(2):1098–1105ADSGoogle Scholar
  16. 16.
    Camara T, Ollivier H, Tillich JP (2005) Constructions and performance of classes of quantum LDPC codes. Preprint quant-ph/0502086Google Scholar
  17. 17.
    Chau HF (1998) Quantum convolutional codes. Phys Rev A 58(2):905–909MathSciNetADSGoogle Scholar
  18. 18.
    Cleve R, Gottesman D (1997) Efficient computations of encodings for quantum error correction. Phys Rev A 56(1):76–82ADSGoogle Scholar
  19. 19.
    Cross AW, DiVincenzo DP, Terhal BM (2007) A comparative code study for quantum fault tolerance. Preprint arXiv:0711.1556 [quant-ph]Google Scholar
  20. 20.
    Dennis E, Kitaev A, Landahl A, Preskill J (2002) Topological quantum memory. J Math Phys 43:4452MathSciNetADSMATHGoogle Scholar
  21. 21.
    DiVincenzo DP (2000) The physical implementation of quantum computation. Fortschr Phys 48(9–11):771–783MATHGoogle Scholar
  22. 22.
    Duan LM, Guo CC (1997) Preserving coherence in quantum computation by pairing quantum bits. Phys Rev Lett 79(10):1953–1956ADSGoogle Scholar
  23. 23.
    Elnozahy EN, Alvisi L, Wang YM, Johnson DB (2002) A survey of rollback-recovery protocols in message-passing systems. ACM Comput Surv 34(3):375–408Google Scholar
  24. 24.
    Forney GD Jr, Grassl M, Guha S (2007) Convolutional and tail-biting quantum error-correcting codes. IEEE Trans Inf Theory 53(3):865–880MathSciNetMATHGoogle Scholar
  25. 25.
    Gottesman D (1996) A class of quantum error-correcting codes saturating the quantum hamming bound. Phys Rev A 54(3):1862–1868MathSciNetADSGoogle Scholar
  26. 26.
    Gottesman D (1997) Stabilizer codes and quantum error correction. Ph?D thesis, California Institute of Technology, PasadenaGoogle Scholar
  27. 27.
    Gottesman D (1998) A theory of fault-tolerant quantum computation. Phys Rev A 57(1):127–137MathSciNetADSGoogle Scholar
  28. 28.
    Gottesman D (1999) Fault-tolerant quantum computation with higher-dimensional systems. Chaos, Solitons & Fractals 10(10):1749–1758MathSciNetMATHGoogle Scholar
  29. 29.
    Gottesman D (2005) Requirements and desiderata for fault-tolerant quantum computing: Beyond the DiVincenzo criteria. http://www.perimeterinstitute.ca/personal/dgottesman/FTreqs.ppt
  30. 30.
    Gottesman D, Chuang IL (1999) Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature 402:390–393ADSGoogle Scholar
  31. 31.
    Grassl M (2002) Algorithmic aspects of quantum error-correcting codes. In: Brylinski RK, Chen G (eds) Mathematics of quantum computation. CRC, Boca Raton, pp 223–252Google Scholar
  32. 32.
    Grassl M, Beth T (2000) Cyclic quantum error-correcting codes and quantum shift registers. Proc Royal Soc London A 456(2003):2689–2706MathSciNetADSMATHGoogle Scholar
  33. 33.
    Grassl M, Rötteler M (2006) Non-catastrophic encoders and encoder inverses for quantum convolutional codes. In: Proceedings 2006 IEEE International Symposium on Information Theory (ISIT 2006), Seattle, pp 1109–1113Google Scholar
  34. 34.
    Grassl M, Rötteler M (2008) Quantum Goethals-Preparata codes. In: Proceedings 2008 IEEE International Symposium on Information Theory (ISIT 2008), Toronto, pp 300–304Google Scholar
  35. 35.
    Grassl M, Beth T, Pellizzari T (1997) Codes for the quantum erasure channel. Phys Rev A 56(1):33–38MathSciNetADSGoogle Scholar
  36. 36.
    Grassl M, Beth T, Rötteler M (2004) On optimal quantum codes. Int J Quantum Inf 2(1):55–64Google Scholar
  37. 37.
    Grassl M, Rötteler M, Beth T (2003) Efficient quantum circuits for non-qubit quantum error-correcting codes. Int J Found Comput Sci 14(5):757–775Google Scholar
  38. 38.
    Hamming RW (1986) Coding and information theory. Prentice-Hall, Englewood CliffsMATHGoogle Scholar
  39. 39.
    Haroche S, Raimond JM (1996) Quantum computing: Dream or nightmare? Phys Today 49(8):51–52Google Scholar
  40. 40.
    Kempe J, Regev O, Unger F, de Wolf R (2008) Upper bounds on the noise threshold for fault-tolerant quantum computing. Preprint arXiv:0802.1464 [quant-ph]Google Scholar
  41. 41.
    Ketkar A, Klappenecker A, Kumar S, Sarvepalli PK (2006) Nonbinary stabilizer codes over finite fields. IEEE Trans Inf Theory 52(11):4892–4914MathSciNetMATHGoogle Scholar
  42. 42.
    Kitaev A (1997) Quantum computations: Algorithms and error correction. Russ Math Surv 52(6):1191–1249MathSciNetMATHGoogle Scholar
  43. 43.
    Knill E (2005) Quantum computing with realistically noisy devices. Nature 434:39–44ADSGoogle Scholar
  44. 44.
    Knill E, Laflamme R (1997) Theory of quantum error-correcting codes. Phys Rev A 55(2):900–911MathSciNetADSGoogle Scholar
  45. 45.
    Knill E, Laflamme R, Zurek WH (1998) Resilient quantum computation: Error models and thresholds. Proc Royal Soc London Series A, 454:365–384. Preprint quant-ph/9702058Google Scholar
  46. 46.
    Knill E, Laflamme R, Milburn G (2001) A scheme for efficient quantum computation with linear optics. Nature 409:46–52ADSGoogle Scholar
  47. 47.
    Kraus K (1983) States, effects, and operations. Lecture notes in physics, vol 190. Springer, BerlinGoogle Scholar
  48. 48.
    Kribs DW, Laflamme R, Poulin D (2005) Unified and generalized approach to quantum error correction. Phys Rev Lett 94(18):180501ADSGoogle Scholar
  49. 49.
    Kribs DW, Laflamme R, Poulin D, Lesosky M (2006) Operator quantum error correction. Quantum Inf Comput 6(3–4):382–399MathSciNetMATHGoogle Scholar
  50. 50.
    Lidar DA, Whaley KB (2003) Decoherence-free subspaces and subsystems. In: Benatti F, Floreanini R (eds) Irreversible quantum dynamics, Lecture Notes in Physics, vol 622. Springer, Berlin, pp 83–120Google Scholar
  51. 51.
    Lidar DA, Chuang IL, Whaley KB (1998) Decoherence-free subspaces for quantum computation. Phys Rev Lett 81(12):2594–2597ADSGoogle Scholar
  52. 52.
    MacKay DJC, Mitchison G, McFadden PL (2004) Quantum computations: Algorithms and error correction. IEEE Trans Inf Theory 50(10):2315–2330MathSciNetGoogle Scholar
  53. 53.
    MacWilliams FJ, Sloane NJA (1977) The theory of error-correcting codes. North-Holland, AmsterdamMATHGoogle Scholar
  54. 54.
    Matsumoto R (2002) Improvement of Ashikhmin–Litsyn–Tsfasman bound for quantum codes. IEEE Trans Inf Theory 48(7):2122–2124MATHGoogle Scholar
  55. 55.
    von Neumann J (1956) Probabilistic logic and the synthesis of reliable organisms from unreliable components. In: Shannon CE, McCarthy J (eds) Automata studies. Princeton University Press, Princeton, pp 43–98Google Scholar
  56. 56.
    Nielsen MA, Chuang IL (2000) Quantum computation and quantum information. Cambridge University Press, CambridgeMATHGoogle Scholar
  57. 57.
    Ollivier H, Tillich JP (2003) Description of a quantum convolutional code. Phys Rev Lett 91(17):177902ADSGoogle Scholar
  58. 58.
    Preskill J (1998) Reliable quantum computers. Proc Royal Soc London A, 454:385–410. Preprint quant-ph/9705031MathSciNetADSMATHGoogle Scholar
  59. 59.
    Rains EM, Hardin RH, Shor PW, Sloane NJA (1997) Nonadditive quantum code. Phys Rev Lett 79(5):953–954ADSGoogle Scholar
  60. 60.
    Razborov A (2004) An upper bound on the threshold quantum decoherence rate. Quantum Inf Comput 4(3):222–228MathSciNetMATHGoogle Scholar
  61. 61.
    Reichardt BW (2006) Error-detection-based quantum fault tolerance against discrete Pauli noise. Ph?D thesis, University of California, BerkeleyGoogle Scholar
  62. 62.
    Reichardt BW (2006) Fault-tolerance threshold for a distance-three quantum code. In: Proceedings of the 2006 International Colloquium on Automata, Languages and Programming (ICALP'06). Lecture Notes in Computer Science, vol 4051. Springer, Berlin, pp 50–61Google Scholar
  63. 63.
    Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423, 623–656, http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html MathSciNetGoogle Scholar
  64. 64.
    Shor PW (1995) Scheme for reducing decoherence in quantum computer memory. Phys Rev A 52(4):R2493–R2496ADSGoogle Scholar
  65. 65.
    Shor PW (1996) Fault-tolerant quantum computation. In: Proceedings of 35th Annual Symposium on Fundamentals of Computer Science (FOCS'96). IEEE Press, Burlington, pp 56–65. Preprint quant-ph/9605011Google Scholar
  66. 66.
    Steane AM (1996) Error correcting codes in quantum theory. Phys Rev Lett 77(5):793–797MathSciNetADSMATHGoogle Scholar
  67. 67.
    Steane AM (1996) Simple quantum error correcting codes. Phys Rev A 54(6):4741–4751MathSciNetADSGoogle Scholar
  68. 68.
    Steane AM (1997) Active stabilization, quantum computation and quantum state synthesis. Phys Rev Lett 78(11):2252–2255ADSGoogle Scholar
  69. 69.
    Wootters WK, Zurek WH (1982) A single quantum cannot be cloned. Nature 299(5886):802–803ADSGoogle Scholar
  70. 70.
    Zalka C (1996) Threshold estimate for fault tolerant quantum computation. Preprint quant-ph/9612028Google Scholar
  71. 71.
    Zanardi P, Rasetti M (1997) Noiseless quantum codes. Phys Rev Lett 79(17):3306–3309ADSGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Markus Grassl
    • 1
  • Martin Rötteler
    • 2
  1. 1.Institute for Quantum Optics and Quantum InformationAustrian Academy of SciencesInnsbruckAustria
  2. 2.NEC Laboratories America, Inc.PrincetonUSA