Quantum Information Processing

, Volume 15, Issue 5, pp 1921–1936 | Cite as

Entanglement-assisted operator codeword stabilized quantum codes

  • Jeonghwan Shin
  • Jun HeoEmail author
  • Todd A. Brun


In this paper, we introduce a unified framework to construct entanglement-assisted quantum error-correcting codes (QECCs), including additive and nonadditive codes, based on the codeword stabilized (CWS) framework on subsystems. The CWS framework is a scheme to construct QECCs, including both additive and nonadditive codes, and gives a method to construct a QECC from a classical error-correcting code in standard form. Entangled pairs of qubits (ebits) can be used to improve capacity of quantum error correction. In addition, it gives a method to overcome the dual-containing constraint. Operator quantum error correction (OQEC) gives a general framework to construct QECCs. We construct OQEC codes with ebits based on the CWS framework. This new scheme, entanglement-assisted operator codeword stabilized (EAOCWS) quantum codes, is the most general framework we know of to construct both additive and nonadditive codes from classical error-correcting codes. We describe the formalism of our scheme, demonstrate the construction with examples, and give several EAOCWS codes


Quantum information Quantum error correction Codeword stabilized quantum codes Entanglement-assisted quantum error-correcting codes Operator quantum error-correcting codes 



TAB would like to thank Ching-Yi Lai and Mark Wilde for useful conversations. TAB acknowledges financial support from NSF Grant CCF-0830801. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2011959). This research was supported by the Ministry of Science, ICT and Future Planning (MSIP), Korea, under the Information Technology Research Center (ITRC) support program (IITP-2015-R0992-15-1017) supervised by the Institute for Information & communications Technology Promotion (IITP).


  1. 1.
    Bacon, D.: Operator quantum error-correcting subsystems for self-correcting quantum memories. Phys. Rev. A 73(1), 012,340 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bowen, G.: Entanglement required in achieving entanglement-assisted channel capacities. Phys. Rev. A 66(5), 052,313 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brun, T.A., Devetak, I., Hsieh, M.H.: Correcting quantum errors with entanglement. Science 314(5798), 436–439 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Calderbank, A., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54(2), 1098–1105 (1996)ADSCrossRefGoogle Scholar
  5. 5.
    Cross, A., Smith, G., Smolin, J.A., Zeng, B.: Codeword stabilized quantum codes. IEEE Trans. Inf. Theory 55(1), 433–438 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gottesman, D.: Stabilizer codes and quantum error correction. Ph.D. thesis, Caltech Ph.D. dissertation, Psadena, CA (1997)Google Scholar
  7. 7.
    Hsieh, M.H., Devetak, I., Brun, T.A.: General entanglement-assisted quantum error-correcting codes. Phys. Rev. A 76(6), 062,313 (2007)CrossRefGoogle Scholar
  8. 8.
    Kribs, D., Laflamme, R., Poulin, D.: Unified and generalized approach to quantum error correction. Phys. Rev. Lett 94(18), 180,501 (2005)CrossRefGoogle Scholar
  9. 9.
    Lai, C.Y., Brun, T.A.: Entanglement increases the error-correcting ability of quantum error-correcting codes. Arxiv preprint arXiv:1008.2598 (2010)
  10. 10.
    Poulin, D.: Stabilizer formalism for operator quantum error correction. Phys. Rev. Lett. 95(23), 230,504 (2005)CrossRefGoogle Scholar
  11. 11.
    Shin, J., Heo, J., Brun, T.A.: Entanglement-assisted codeword stabilized quantum codes. Phys. Rev. A 84, 062,321 (2011)CrossRefGoogle Scholar
  12. 12.
    Shin, J., Heo, J., Brun, T.A.: Codeword-stabilized quantum codes on subsystems. Phys. Rev. A 86, 042,318 (2012)CrossRefGoogle Scholar
  13. 13.
    Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52(4), R2493–R2496 (1995)ADSCrossRefGoogle Scholar
  14. 14.
    Steane, A.M.: Error Correcting Codes in Quantum Theory. Phys. Rev. Lett. 77, 793–797 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Van den Nest, M., Dehaene, J., De Moor, B.: Graphical description of the action of local Clifford transformations on graph states. Phys. Rev. A 69(2), 022,316 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of Electrical EngineeringKorea UniversitySeoulKorea
  2. 2.Communication Sciences InstituteUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations