Skip to main content
SpringerLink
Log in

On the Construction of Asymmetric Quantum Codes

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

Several families of good nonbinary asymmetric quantum codes are constructed in this paper. These new quantum codes are derived from the Calderbank-Shor-Steane (CSS) construction as well as the Hermitian construction applied respectively to two classical nested Bose-Chaudhuri-Hocquenghem (BCH) codes where one of them are additionally Euclidean (Hermitian) dual-containing. The asymmetric codes constructed here have parameters better than the ones available in the literature.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aly, S.A.: Asymmetric quantum BCH codes. In: Proc. IEEE International Conference on Computer Engineering and Systems (ICCES’08), pp. 157–162 (2008)

    Chapter  Google Scholar 

  2. Aly, S.A., Ashikhmin, A.: Nonbinary quantum cyclic and subsystem codes over asymmetrically-decohered quantum channels. E-print. arXiv:1002.2966 [quant-ph]

  3. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans. Inf. Theory 53(3), 1183–1188 (2007)

    Article  MathSciNet  Google Scholar 

  4. Ashikhmin, A., Knill, E.: Non-binary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54(2), 1098–1106 (1996)

    Article  ADS  Google Scholar 

  6. Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44(4), 1369–1387 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen, J.-Z., Li, J.-P., Lin, J.: New optimal asymmetric quantum codes derived from negacyclic codes. Int. J. Theor. Phys. (2014). doi:10.1007/s10773-013-1784-z

    MathSciNet  Google Scholar 

  8. Evans, Z.W.E., Stephens, A.M., Cole, J.H., Hollenberg, L.C.L.: Error correction optimisation in the presence of x/z asymmetry. E-print. arXiv:0709.3875 [quant-ph]

  9. Ezerman, M.F., Jitman, S., Ling, S.: On asymmetric quantum MDS codes. e-print. arXiv:1006.1694 [quant-ph]

  10. Ezerman, M.F., Ling, S., Solé, P.: Additive asymmetric quantum codes. E-print. arXiv:1002.4088 [quant-ph]

  11. Hamada, M.: Concatenated quantum codes constructible in polynomial time: efficient decoding and error correction. IEEE Trans. Inf. Theory 54(12), 5689–5704 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ioffe, L., Mezard, M.: Asymmetric quantum error-correcting codes. Phys. Rev. A 75, 032345 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  13. Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4892–4914 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. La Guardia, G.G.: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80(4), 042331 (2009), 11 pp.

    Article  ADS  Google Scholar 

  15. La Guardia, G.G.: New families of asymmetric quantum BCH codes. Quantum Inf. Comput. 11(3–4), 239–252 (2011)

    MATH  MathSciNet  Google Scholar 

  16. La Guardia, G.G.: Asymmetric quantum Reed-Solomon and generalized Reed-Solomon codes. Quantum Inf. Process. 11, 591–604 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  17. La Guardia, G.G.: Asymmetric quantum product codes. Int. J. Quantum Inf. 10(1), 1250005 (2012)

    Article  MathSciNet  Google Scholar 

  18. La Guardia, G.G.: Asymmetric quantum codes: new codes from old. Quantum Inf. Process. 12, 2771–2790 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. La Guardia, G.G., Palazzo, R.: Jr.. Constructions of new families of nonbinary CSS codes. Discrete Math. 310(21), 2935–2945 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Li, R., Zuo, F., Liu, Y., Xu, Z.: Hermitian dual containing BCH codes and construction of new quantum codes. Quantum Inf. Comput. 13(1–2), 21–35 (2013)

    MathSciNet  Google Scholar 

  21. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)

    MATH  Google Scholar 

  22. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  23. Qian, J., Zhang, L.: New optimal subsystem codes. Discrete Math. 313, 2451–2455 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  24. Qian, J., Zhang, L.: New optimal asymmetric quantum codes. Mod. Phys. Lett. B 27(2), 1350010 (2013), 8 pp.

    Article  ADS  MathSciNet  Google Scholar 

  25. Riguang, L., Zhi, M.: Constructions of new families of nonbinary asymmetric quantum BCH codes and subsystem BCH codes. Sci. China, Phys. Mech. Astron. 55(3), 465–469 (2012)

    Google Scholar 

  26. Sarvepalli, P.K., Klappenecker, A., Rötteler, M.: Asymmetric quantum LDPC codes. In: Proc. Int. Symp. Inform. Theory (ISIT), pp. 6–11 (2008)

    Google Scholar 

  27. Sarvepalli, P.K., Klappenecker, A., Rötteler, M.: Asymmetric quantum codes: constructions, bounds and performance. In: Proc. of the Royal Society A, pp. 1645–1672 (2009)

    Google Scholar 

  28. Steane, A.M.: Simple quantum error correcting-codes. Phys. Rev. A 54, 4741–4751 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  29. Steane, A.M.: Enlargement of Calderbank-Shor-Steane quantum codes. IEEE Trans. Inf. Theory 45(7), 2492–2495 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  30. Stephens, A.M., Evans, Z.W.E., Devitt, S.J., Hollenberg, L.C.L.: Asymmetric quantum error correction via code conversion. Phys. Rev. A 77, 062335 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  31. Wang, L., Feng, K., Ling, S., Xing, C.: Asymmetric quantum codes: characterization and constructions. IEEE Trans. Inf. Theory 56(6), 2938–2945 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was partially supported by the Brazilian agencies CAPES and CNPq.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, State University of Ponta Grossa (UEPG), 84030-900, Ponta Grossa, PR, Brazil

    Giuliano G. La Guardia

Authors
  1. Giuliano G. La Guardia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giuliano G. La Guardia.

Rights and permissions

Reprints and permissions

About this article

Cite this article

La Guardia, G.G. On the Construction of Asymmetric Quantum Codes. Int J Theor Phys 53, 2312–2322 (2014). https://doi.org/10.1007/s10773-014-2031-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-014-2031-y

Keywords

Search

Navigation