Skip to main content
SpringerLink
Log in

Three classes of new EAQEC MDS codes

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Entanglement-assisted quantum error-correcting (EAQEC) codes can be derived from arbitrary classical linear codes. However, it is a very difficult task to determine the number c of pre-shared maximally entangled states. In this paper, we first give a new formula for calculating the number c of pre-shared maximally entangled states. Then, using this formula, we construct three classes of new entanglement-assisted quantum error-correcting maximum-distance-separable (EAQEC MDS) codes. In addition, our obtained EAQEC MDS codes 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.

Similar content being viewed by others

References

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. Abdel-Aty, A.H., Kadry, H., Zidan, M., Al-Sboug, Y., Zanaty, E., Abdel-Aty, A.M.: A quantum classification algorithm for classification incomplete patterns based on entanglement measure. J. Intell. Fuzzy Syst. 38(3), 2809–2816 (2020)

    Google Scholar 

  4. Bowen, G.: Entanglement required in achieving entanglement-assisted channel capacities. Phys. Rev. A 66, 052313 (2002)

    ADS  Google Scholar 

  5. Brun, T., Devetak, I., Hsieh, M.H.: Correcting quantum errors with entanglement. Science 314, 436–439 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  7. Chen, J., Huang, Y., Feng, C., Chen, R.: Entanglement-assisted quantum MDS codes constructed from negacyclic codes. Quantum Inf. Process. 16(303), 1–22 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  8. Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61, 1474–1484 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Ding, H., Wu, J., Li, X.: Evolving neural network using hybrid genetic algorithm and simulated annealing for rainfall runoff forecasting. Adv. Swarm Intell. 7331, 444–451 (2011)

    Google Scholar 

  10. Dunjko, V., Briegel, H.J.: Machine learning and artificial intelligence in the quantum domain: a review of recent progress. Rep. Prog. Phys. 81, 074001 (2018)

    ADS  MathSciNet  Google Scholar 

  11. Fujiwara, Y., Clark, D., Vandendriessche, P., De Boeck, M., Tonchev, V.D.: Entanglement-assisted quantum low-density parity-check codes. Phys. Rev. A 82, 042338 (2010)

    ADS  Google Scholar 

  12. Fan, J., Chen, H., Xu, J.: Construction of q-ary entanglement-assisted quantum MDS codes with minmum distanc greater than q+1. Quantum Inf. Comput. 16, 423–434 (2016)

    MathSciNet  Google Scholar 

  13. Fan, Y., Zhang, L.: Galois self-dual constacyclic codes. Des. Codes Cryptogr. 84, 473–492 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Gabidulin, E.: Theory of codes with maximum rank distance. Probl. Inf. Transm. 1(2), 1–12 (1985)

    MathSciNet  MATH  Google Scholar 

  15. Guenda, K., Gulliver, T.A., Jitman, S., Thipworawimon, S.: Linear l-intersection pairs of codes and their applications. Des. Codes Cryptogr. 88(1), 133–152 (2020)

    MathSciNet  MATH  Google Scholar 

  16. Hurley, T., Hurley, D.: Coding theory: the unit-derived methodology. Int. J. Inf. Coding Theory 5(1), 55–80 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Jin, L., Xing, C.: New MDS self-dual codes from generalized Reed–Solomon codes. IEEE Trans. Inf. Theory 63, 1434–1438 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Jin, L., Ling, S., Luo, J., Xing, C.: Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes. IEEE Trans. Inf. Theory 56, 4735–4740 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inf. Theory 59, 1193–1197 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Ketkar, A., Klappenecker, A., Kumar, S.: Nonbinary stablizer codes over finite fields. IEEE Trans. Inf. Theory 52, 4892–4914 (2006)

    MATH  Google Scholar 

  21. Kshevelskiy, A., Gabidulin, E.: The new construction of rank code. Probl. Inf. Transm. 1(2), 2105–2108 (2005)

    Google Scholar 

  22. Koroglu, M.E.: New entanglement-assisted MDS quantum codes from constacyclic codes. Quantum Inf. Process. 18, 44 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  23. Liu, Y., Li, R., Lv, L., Ma, Y.: A class of constacyclic BCH codes and new quantum codes. Quantum Inf. Process. 16, 66 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  24. Lai, C.Y., Brun, T.A., Wilde, M.M.: Dualityinentanglement-assisted quantum error correction. IEEE Trans. Inf. Theory 59, 4020–4024 (2013)

    MATH  Google Scholar 

  25. Lai, C.Y., Brun, T.A.: Entanglement increases the error-correcting ability of quantum error-correcting codes. Phys. Rev. A 88, 012320 (2013)

    ADS  Google Scholar 

  26. Luo, L., Ma, Z., Wei, Z., Leng, R.: Non-binary entanglement-assisted quantum stabilizer codes. Sci. China Inf. Sci. 60, 04250:11-042501:14 (2017)

    Google Scholar 

  27. Lu, L., Li, R., Guo, L., Ma, Y., Liu, Y.: Entanglement-assisted quantum MDS codes from negacyclic codes. Quant. Inf. Process. 17, 69 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  28. Lu, L., Ma, W., Li, R., Ma, Y., Liu, Y., Cao, H.: Entanglement-assisted quantum MDS codes from constacyclic codes with large minimum distance. Finite Fields Appl. 53, 309–325 (2018)

    MathSciNet  MATH  Google Scholar 

  29. Liu, X., Yu, L., Hu, P.: New entanglement-assisted quantum codes from k-Galois dual codes. Finite Fields Appl. 55, 21–32 (2019)

    MathSciNet  MATH  Google Scholar 

  30. Liu, X., Liu, H., Yu, L.: New EAQEC codes constructed from Galois LCD codes. Quant. Inf. Process. 19, 20 (2020)

    ADS  MathSciNet  Google Scholar 

  31. Luo, G., Cao, X.: Two new families of entanglement-assisted quantum MDS codes from generalized Reed–Solomon codes. Quant. Inf. Process. 18, 89 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  32. Macwilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland Publishing Company, Amsterdam (1977)

    MATH  Google Scholar 

  33. Mustafa, S., Emre, K.: An application of constacyclic codes to entanglement-assisted quantum MDS codes. Comput. Appl. Math. 38(75), 1–13 (2019)

    MathSciNet  MATH  Google Scholar 

  34. Pereira, F. R. F., Entanglement-assisted quantum codes from cyclic codes. arXiv:1911.0638v1

  35. Qian, J., Zhang, L.: On MDS linear complementary dual codes and entanglement-assisted quantum codes. Des. Codes Cryptogr. 86(7), 1565–1572 (2018)

    MathSciNet  MATH  Google Scholar 

  36. Qian, J., Zhang, L.: Constructions of new entanglement-assisted quantum MDS and almost MDS codes. Quantum Inf. Process. 18, 71 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  37. Sagheer, A., Zidan, M., Abdelsamea, M.M.: A novel autonomous perceptron model for pattern classification applications. Entropy 21(8), 763 (2019)

    ADS  MathSciNet  Google Scholar 

  38. Wang, J., Li, R., Lv, J., Guo, G., Liu, Y.: Entanglement-assisted quantum error correction codes with length \(n=q^2+1\). Quant. Inf. Proces. 18, 292 (2019)

    ADS  Google Scholar 

  39. Wilde, M.M., Brun, T.A.: Optimal entanglement formulas for entanglement-assisted quantum coding. Phys. Rev. A 77, 064302 (2008)

    ADS  Google Scholar 

  40. Zidan, M., Sagheer, A., Metwally, N.: An autonomous competitive learning algorithm using quantum hamming neural networks. In: Proceedings of the International Joint Conference on Neural Networks (IJCNN, Killarney, Ireland, 2015), pp. 1–7. IEEE (2015)

  41. Zidan, M., Abdel-Aty, A.H., El-Sadek, A.E., Zanaty, A., Abdel-Aty, A.M.: Low-cost autonomous prceptron neural network inspired by quantum computation. AIP Conf. Proc. 1905(1), 020005 (2017)

    Google Scholar 

  42. Zidan, M., Abdel-Aty, A.H., Younes, A., Zanaty, E.A., El-khayat, I., Abdel-Aty, A.M.: A novel algorithm based on entanglement measurement for improving speed of quantum algorithms. Appl. Math. Inf. Sci. 12(1), 265–269 (2018)

    MathSciNet  Google Scholar 

  43. Zidan, M., Abdel-Aty, A.H., Nguyene, D.M., Mohamed, A.S.A., Al-Sboug, Y., Eleuch, H., Abdel-Aty, A.M.: A quantum algorithm based on entanglement measure for classifying multivariate function into novel hidden classes. Results Phys. 15, 102549 (2019)

    Google Scholar 

  44. Zidan, M., Abdel-Aty, A.H., El-shafei, M., Feraig, M., Al-Sbou, Y., Eleuch, H., Abdel-Aty, A.M.: Quantum classification algorithm based on competitive learning neural network and entanglement measure. Appl. Sci. 9(7), 1277 (2019)

    Google Scholar 

  45. Zidan, M.: A novel quantum computing model based on entanglement degree. Mod. Phys. Lett. B. 1, 2050401 (2020). https://doi.org/10.1142/S0217984920504011

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by Scientific Research Foundation of Hubei Provincial Education Department of China. (Grant No. Q20174503) and the National Science Foundation of Hubei Polytechnic University of China (Grant No. 17xjz03A).

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Hubei Polytechnic University, Huangshi, 435003, Hubei, China

    Peng Hu & Xiusheng Liu

  2. School of Science and Technology, College of Arts and Science of Hubei Normal University, Huangshi, 435109, Hubei, China

    Xiusheng Liu

Authors
  1. Peng Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiusheng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peng Hu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, P., Liu, X. Three classes of new EAQEC MDS codes. Quantum Inf Process 20, 103 (2021). https://doi.org/10.1007/s11128-021-03039-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-021-03039-7

Keywords

Search

Navigation