Quantum Information Processing

, Volume 4, Issue 5, pp 399–431 | Cite as

Probabilities of Failure for Quantum Error Correction

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Abstract

We investigate the performance of a quantum error-correcting code when pushed beyond its intended capacity to protect against errors, presenting formulae for the probability of failure when the errors affect more qudits than that specified by the code’s minimum distance. Such formulae provide a means to rank different codes of the same minimum distance. We consider both error detection and error correction, treating explicit examples in the case of stabilizer codes constructed from qubits and encoding a single qubit

Keywords

Quantum error correction quantum information 

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References

  1. 1.
    Calderbank A.R., Rains E.M., Shor P.W., Sloane N.J.A. (1998). IEEE Trans. Inform. Theory 44:1369MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Gottesman D., PhD Thesis, California Institute of Technology, Pasadena CA, (1997); e-print quant-ph/9705052Google Scholar
  3. 3.
    Knill E., Laflamme R. (1997). Phys. Rev. A 55:900CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Preskill J., Lecture Notes for Physics 219: Quantum Computation (California Institute of Technology, Pasadena CA, 1998). URL: http://www.theory.caltech.edu/people/preskill/ph219/Google Scholar
  5. 5.
    Nielsen M.A., Chuang I.L. (2000). Quantum Computation and Quantum Information. Cambridge University Press, CambridgeMATHGoogle Scholar
  6. 6.
    Grassl M., in Mathematics of Quantum Computation:edited by R. K. Brylinski and G. Chen (Chapman & Hall/CRC, London, 2002Google Scholar
  7. 7.
    Shor P. (1995). Phys. Rev. A 52:2493CrossRefADSGoogle Scholar
  8. 8.
    Steane A.M. (1996). Phys. Rev. Lett 77:793MATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Ashikhmin A.E., Barg A.M., Knill E., Litsyn S.N. (2000). IEEE Trans. Inform. Theory 46:778MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Rains E.M. (1998). IEEE Trans. Inform. Theory 44:1388MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Shor P., Laflamme R. (1997). Phys. Rev. Lett 78:1600CrossRefADSGoogle Scholar
  12. 12.
    Rains E.M. (1999). IEEE Trans. Inform. Theory 45:1827MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ashikhmin A., Knill E. (2001). IEEE Trans. Inform. Theory 47:3065MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Grassl M., Beth T., Rötteler M. (2004). Int. J. Quantum Inf 2:55MATHCrossRefGoogle Scholar
  15. 15.
    Gaborit P., Huffman W.C., Kim J.-L., Pless V. in DIMACS Series in Discrete Mathematics and Theoretical Computer Science Volume 56: Codes and Association Schemes:edited by A. Barg and S. Litsyn (American Mathematical Society, Providence RI, 2001Google Scholar
  16. 16.
    Bachoc C., Gaborit P. Journal de Théorie des Nombres de Bordeaux 12:255 (2000); Electronic Notes in Discrete Mathematics 6:(2001).Google Scholar

Copyright information

© Springer Science+Business Media Inc 2005

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity of New MexicoAlbuquerqueUSA

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