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Quantum Codes Do Not Fix Qubit Independent Errors

Received: 30 May 2021     Accepted: 15 July 2021     Published: 31 August 2021
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Abstract

The goal of this work is to analyse the performance of quantum error correction codes in regard to fixing independent errors in several qubits. We say that a quantum code does not fix a quantum computing error if its application does not reduce the variance of the error. We restrict our study to the classical 5−qubit quantum error correcting code (proposed by Laflamme and some collaborators), which is able to correct arbitrary errors in a single qubit and is also fault-tolerant. We show that this code does not fix qubit independent errors, even assuming that the correction circuit does not introduce new errors. We also prove, for qubit independent errors, that if the correction circuit of the 5−qubit quantum code detects an error, then the corrected state has central symmetry and, as a consequence, its variance is maximal. We have been able to obtain these results thanks to the high symmetry of the 5−qubit quantum code. Although the calculations needed to extend our proofs to less symmetric codes seem to be extremely complicated, we nevertheless think that the results obtained for the 5−qubit quantum code reveal a general behavior pattern of quantum error correcting codes against qubit independent errors.

Published in American Journal of Information Science and Technology (Volume 5, Issue 3)
DOI 10.11648/j.ajist.20210503.12
Page(s) 60-72
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

5−Qubit Quantum Code, Quantum Error Correcting Codes, Qubit Independent Quantum Computing Errors, Quantum Computing Error Variance

References
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[2] Calderbank, A. R., Shor, P. W., Good quantum error- correcting codes exist, Phys. Rev. A 54, 1098–1105 (1996).
[3] Steane, A. M., Multiple particle inference and quantum error correction, Proc. Roy. Soc. A. 452, 2551 (1996).
[4] Gottesman, D., Stabilizer Codes and Quantum Error Correction. PhD thesis, California Institute of Technology, 1997.
[5] Calderbank, A. R., Rains, E. M., Shor, P. W., Sloane, N. J. A., Fellow, Quantum error correction via codes over GF (4), IEEE Trans. Inf. Theory 44 (4), 1369–1387 (1998).
[6] Shor, P. W., Fault-tolerant quantum computation. Symposium on the Foundations of Computer Science, Los Alamitos, CA, (1996).
[7] Preskill, J., Reliable quantum computers, Proc. Roy. Soc. Lond. A 454, 385–410 (1998).
[8] Steane, A. M., Active stabilization, quantum computation and quantum state synthesis, Phys. Rev. Lett. 78, 2252 (1997).
[9] Gottesman, D., Theory of fault-tolerant quantum computation, Phys. Rev. A 57, 127–137 (1998).
[10] Knill, E., Laflamme, R., Zurek, W. H., Resilient Quantum Computation, Science 279, 5349 (1998). arXiv: quant- ph/9702058v1. DOI: 10.1126/science.279.5349.342
[11] Kitaev, A. Yu., Fault-tolerant quantum computation by anyons, Annals of Physics 303 (1) 2–30 (2003). arXiv: quant-ph/9707021. DOI: 10.1016/S0003- 4916(02)00018-0
[12] Aharonov, D., Ben-Or, M., Fault-Tolerant Quantum Computation with Constant Error Rate. SIAM Journal on Computing 38 (4): 1207–1282 (2008). arXiv: quant- ph/9906129. DOI: 10.1137/S0097539799359385
[13] Bennet, C. H., DiVincenzo, D. P., Smolin, J. A., Wootters, W. K., Mixed state entanglement and quantum error correction, Los Alamos Physics Preprint Archive http://xxx.lanl.gov/abs/quant-ph/9909058, (1999).
[14] Laflamme, R., Miquel, C., Paz, J.-P., Zurek, W. H., Perfect quantum error correction codes, Phys. Rev. Lett. 77, 198 (1996). Los Alamos Physics Preprint Archive http://xxx.lanl.gov/abs/quant-ph/9602019.
[15] Nielsen, M. A., Chuang, I. L., Quantum Computation and Quantum Information, Cambridge University Press, 2010.
[16] Lacalle, J., Pozo Coronado, L. M., Variance of the sum of independent quantum computing errors, Quantum Information & Computation 19 (15-16), 1294–1312 (2019).
[17] Lacalle, J., Pozo Coronado, L. M., Fonseca de Oliveira, A. L., Quantum codes do not fix isotropic errors, Quantum Information Processing, accepted (2020).
[18] Lacalle, J., Pozo Coronado, L. M., Fonseca de Oliveira, A. L., Mart´ın-Cuevas, R., Quantum codes do not fix qubit independent errors - Monitoring of calculations, https://github.com/rafamartinc/quantum codes do not fix qubit independent errors
[19] Shor, P.W., Schemeforreducingdecoherenceinquantum computer memory, Phys. Rev. A 52, R2493–R2496 (1995).
[20] Steane, A., Multiple-Particle Interference and Quantum Error Correction, Proc. Roy. Soc. Lond. A. 452 1954, 2551–2577 (1996).
Cite This Article
  • APA Style

    Jesús Lacalle, Luis Miguel Pozo Coronado, André Luiz Fonseca de Oliveira, Rafael Mart´ın-Cuevas. (2021). Quantum Codes Do Not Fix Qubit Independent Errors. American Journal of Information Science and Technology, 5(3), 60-72. https://doi.org/10.11648/j.ajist.20210503.12

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    ACS Style

    Jesús Lacalle; Luis Miguel Pozo Coronado; André Luiz Fonseca de Oliveira; Rafael Mart´ın-Cuevas. Quantum Codes Do Not Fix Qubit Independent Errors. Am. J. Inf. Sci. Technol. 2021, 5(3), 60-72. doi: 10.11648/j.ajist.20210503.12

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    AMA Style

    Jesús Lacalle, Luis Miguel Pozo Coronado, André Luiz Fonseca de Oliveira, Rafael Mart´ın-Cuevas. Quantum Codes Do Not Fix Qubit Independent Errors. Am J Inf Sci Technol. 2021;5(3):60-72. doi: 10.11648/j.ajist.20210503.12

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  • @article{10.11648/j.ajist.20210503.12,
      author = {Jesús Lacalle and Luis Miguel Pozo Coronado and André Luiz Fonseca de Oliveira and Rafael Mart´ın-Cuevas},
      title = {Quantum Codes Do Not Fix Qubit Independent Errors},
      journal = {American Journal of Information Science and Technology},
      volume = {5},
      number = {3},
      pages = {60-72},
      doi = {10.11648/j.ajist.20210503.12},
      url = {https://doi.org/10.11648/j.ajist.20210503.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajist.20210503.12},
      abstract = {The goal of this work is to analyse the performance of quantum error correction codes in regard to fixing independent errors in several qubits. We say that a quantum code does not fix a quantum computing error if its application does not reduce the variance of the error. We restrict our study to the classical 5−qubit quantum error correcting code (proposed by Laflamme and some collaborators), which is able to correct arbitrary errors in a single qubit and is also fault-tolerant. We show that this code does not fix qubit independent errors, even assuming that the correction circuit does not introduce new errors. We also prove, for qubit independent errors, that if the correction circuit of the 5−qubit quantum code detects an error, then the corrected state has central symmetry and, as a consequence, its variance is maximal. We have been able to obtain these results thanks to the high symmetry of the 5−qubit quantum code. Although the calculations needed to extend our proofs to less symmetric codes seem to be extremely complicated, we nevertheless think that the results obtained for the 5−qubit quantum code reveal a general behavior pattern of quantum error correcting codes against qubit independent errors.},
     year = {2021}
    }
    

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    AU  - Jesús Lacalle
    AU  - Luis Miguel Pozo Coronado
    AU  - André Luiz Fonseca de Oliveira
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    DO  - 10.11648/j.ajist.20210503.12
    T2  - American Journal of Information Science and Technology
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    AB  - The goal of this work is to analyse the performance of quantum error correction codes in regard to fixing independent errors in several qubits. We say that a quantum code does not fix a quantum computing error if its application does not reduce the variance of the error. We restrict our study to the classical 5−qubit quantum error correcting code (proposed by Laflamme and some collaborators), which is able to correct arbitrary errors in a single qubit and is also fault-tolerant. We show that this code does not fix qubit independent errors, even assuming that the correction circuit does not introduce new errors. We also prove, for qubit independent errors, that if the correction circuit of the 5−qubit quantum code detects an error, then the corrected state has central symmetry and, as a consequence, its variance is maximal. We have been able to obtain these results thanks to the high symmetry of the 5−qubit quantum code. Although the calculations needed to extend our proofs to less symmetric codes seem to be extremely complicated, we nevertheless think that the results obtained for the 5−qubit quantum code reveal a general behavior pattern of quantum error correcting codes against qubit independent errors.
    VL  - 5
    IS  - 3
    ER  - 

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Author Information
  • Department of Mathematics Applied to Information and Communications Technologies, Universidad Politécnica de Madrid, Madrid, Spain

  • Department of Mathematics Applied to Information and Communications Technologies, Universidad Politécnica de Madrid, Madrid, Spain

  • School of Engineering, Universidad ORT Uruguay, Montevideo, Uruguay

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