Entanglement-Assisted Quantum Error-Correcting Codes

  • Igor Devetak
  • Todd A. Brun
  • Min-Hsiu Hsieh


We develop the theory of entanglement-assisted quantum error correcting codes (EAQECCs), a generalization of the stabilizer formalism to the setting in which the sender and receiver have access to pre-shared entanglement. Conventional stabilizer codes are equivalent to self-orthogonal symplectic codes. In contrast, EAQECCs do not require self-orthogonality, which greatly simplifies their construction. We show how any classical quaternary block code can be made into a EAQECC. Furthermore, the error-correcting power of the quantum codes follows directly from the power of the classical codes.


Turbo Code Quantum Code Pauli Operator Abelian Extension Classical Code 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    C. Berrou, A. Glavieux, and P. Thitimajshima, Near Shannon limit error correcting coding and decoding: Turbo-codes. In: Proc. ICC’93, Geneva, Switzerland, May 1993, pp. 1064–1070 (1993) Google Scholar
  2. 2.
    G. Bowen, Entanglement required in achieving entanglement-assisted channel capacities. Phys. Rev. A 66, 052313 (2002) CrossRefADSGoogle Scholar
  3. 3.
    S. Bravyi, D. Fattal, and D. Gottesman, GHZ extraction yield for multipartite stabilizer states. J. Math. Phys. 47, 062106 (2006) CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    A.R. Calderbank, and P.W. Shor, Good quantum error-correcting codes exist. Phys. Rev. A 54, 1098–1105 (1996) CrossRefADSGoogle Scholar
  5. 5.
    A.R. Calderbank, E.M. Rains, P.W. Shor, and N.J.A. Sloane, Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44, 1369–1387 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    I. Devetak, A.W. Harrow, and A. Winter, A resource framework for quantum Shannon theory. IEEE Trans. Inf. Theory 54(10), 4587–4618 (2008) CrossRefMathSciNetGoogle Scholar
  7. 7.
    D. Fattal, T.S. Cubitt, Y. Yamamoto, S. Bravyi, and I.L. Chuang, Entanglement in the stabilizer formalism. quant-ph/0406168 (2004)
  8. 8.
    R.G. Gallager, Low-density parity-check codes. PhD thesis, Massachusetts Institute of Technology (1963) Google Scholar
  9. 9.
    D. Gottesman, Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A 54, 1862–1868 (1996) CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    D. Gottesman, Stabilizer codes and quantum error correction. PhD thesis, California Institute of Technology (1997) Google Scholar
  11. 11.
    M.H. Hsieh, T. Brun, and I. Devetak, Entanglement-assisted quantum quasicyclic low-density parity-check codes. Phys. Rev. A 79, 032340 (2009) CrossRefADSGoogle Scholar
  12. 12.
    E. Knill, and R. Laflamme, A theory of quantum error correcting codes. Phys. Rev. A 55, 900–911 (1997) CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    D.J.C. MacKay, G. Mitchison, and P.L. McFadden, Sparse-graph codes for quantum error correction. IEEE Trans. Inf. Theory 50, 2315–2330 (2004) CrossRefMathSciNetGoogle Scholar
  14. 14.
    M.A. Nielsen, and I.L. Chuang, Quantum Computation and Quantum Information. Cambridge University Press, New York (2000) zbMATHGoogle Scholar
  15. 15.
    J. Preskill, Lecture notes for physics 229: Quantum information and computation (1998).
  16. 16.
    P.W. Shor, Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, 2493–2496 (1995) CrossRefADSGoogle Scholar
  17. 17.
    A.M. Steane, Error-correcting codes in quantum theory. Phys. Rev. Lett. 77, 793–797 (1996) zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    W.G. Unruh, Maintaining coherence in quantum computers. Phys. Rev. A 51, 992–997 (1995) CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Ming Hsieh Electrical Engineering DepartmentUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Communication Sciences Institute, Department of Electrical Engineering SystemsUniversity of Southern CaliforniaLos AngelesUSA
  3. 3.Ming Hsieh Electrical Engineering DepartmentUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations