Quantum Codes Do Not Fix Qubit Independent Errors

Jesús Lacalle; Luis Miguel Pozo Coronado; André Luiz Fonseca de Oliveira; Rafael Mart´ın-Cuevas

doi:doi:10.11648/j.ajist.20210503.12

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Quantum Codes Do Not Fix Qubit Independent Errors

Jesús Lacalle, Luis Miguel Pozo Coronado, André Luiz Fonseca de Oliveira, Rafael Mart´ın-Cuevas

Published in American Journal of Information Science and Technology (Volume 5, Issue 3)

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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[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).
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[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
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[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.
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[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
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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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TY  - JOUR
T1  - Quantum Codes Do Not Fix Qubit Independent Errors
AU  - Jesús Lacalle
AU  - Luis Miguel Pozo Coronado
AU  - André Luiz Fonseca de Oliveira
AU  - Rafael Mart´ın-Cuevas
Y1  - 2021/08/31
PY  - 2021
N1  - https://doi.org/10.11648/j.ajist.20210503.12
DO  - 10.11648/j.ajist.20210503.12
T2  - American Journal of Information Science and Technology
JF  - American Journal of Information Science and Technology
JO  - 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

Jesús Lacalle

Department of Mathematics Applied to Information and Communications Technologies, Universidad Politécnica de Madrid, Madrid, Spain
Luis Miguel Pozo Coronado

Department of Mathematics Applied to Information and Communications Technologies, Universidad Politécnica de Madrid, Madrid, Spain
André Luiz Fonseca de Oliveira

School of Engineering, Universidad ORT Uruguay, Montevideo, Uruguay
Rafael Mart´ın-Cuevas

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Plain Text BibTeX RIS

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Quantum Codes Do Not Fix Qubit Independent Errors
AU  - Jesús Lacalle
AU  - Luis Miguel Pozo Coronado
AU  - André Luiz Fonseca de Oliveira
AU  - Rafael Mart´ın-Cuevas
Y1  - 2021/08/31
PY  - 2021
N1  - https://doi.org/10.11648/j.ajist.20210503.12
DO  - 10.11648/j.ajist.20210503.12
T2  - American Journal of Information Science and Technology
JF  - American Journal of Information Science and Technology
JO  - American Journal of Information Science and Technology
SP  - 60
EP  - 72
PB  - Science Publishing Group
SN  - 2640-0588
UR  - https://doi.org/10.11648/j.ajist.20210503.12
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  -

Copy | Download