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Semiclassical grounds of the Calderbank-Shor-Steane quantum error correction codes
Keywords:
Quantum Information, Quantum Computation, Linear classical correction codes, Encoding, Syndrome, Quantum error correction codesAbstract
A valid Calderbank-Shor-Steane (CSS) error correction code requires two classical linear codes for the preparation of the initial state (codewords). This code allow to correct for certain errors caused by an unwanted interaction which produces a degraded quantum state. However, this initial seven qubits encoding can be obtained from a maximally entangled Bell state (0000000>+|1111111)/√ through an operation Hint whose explicit expression is derived in the present work. The price the CSS syndrome has to pay due to its classical grounds is that the operator Hint is not unitary. In other words, Hint is not a valid quantum gate i. e. this does not represent a logical operation. Consequently, the final state is not completely robust for the standard cryptography of Quantum Computation. Besides to be a non unitary operator, Hint, is not reversible introducing with this dissipative effects that destroy the coherence in the quantum computer. Additionally, this operator is not invariant under rotations of the protector qubits inducing then preferred directions of the propagation of the logical information. These are indeed the reasons that prompt us for extending the semi classical CSS quantum error correction codes formalism to a pure quantum Hamming codes.
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References
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[7] P. Shor, “Fault-tolerant quantum computation”, Proceedings of the 37th Symposium on the Foundations of Computer Sciences (FOCS), 1995, 37 Los Alamitos, CA: IEEE Press, pp. 56-57.
[8] A. Y. Kitaev, “Quantum Error Corrections with Imperfect Gates”. In Quantum Communication and Computing and Measurement Edited by O. Hirota et al Plenum, New York.
[9] E. Knill, and R. Laflamme; “Concatenated Quantum Codes”, arxiv:quant-ph/9608012.
[10] D. Aharonov, and M. Ben-Or; “Fault-Tolerant Quantum Computation With Constant Error Rate”, arxiv:quantph/ 9906129.
[11] E. Knill, R. Laflamme, and W. H. Zurek, “Resilient Quantum Computation”, Science, Vol. 279, 1998, pp. 342-345.
[12] E. Knill, R. Laflamme, and W. H. Zurek, “Resilient quantum computation: error models and thresholds”, Proc. R. Soc. London Ser. A, Vol. 454, No. 1969, 1998, pp. 365-384.
[13] D. Gottesman, “Theory of fault-tolerant quantum computation”, Phys. Rev., Vol. A57, 1998, pp. 127-137.
[14] J. Preskill, “ Reliable quantum computers”, Proc. R. Soc. London, Ser. A, Vol. 454, No. 1969, 1998. pp. 385-410.
[15] A. Skander, M. Nadjim, and M. Benslama, “A New Error Correction Intuitive Approach in Quantum Communications Protocols”, Am. J. Appl. Sci., Vol. 4, No. 8, 2007, pp. 597-604.
[16] A. Calderbank, and P. Shor, “Good quantum errorcorrecting codes exist”, Phys. Rev. Vol. A 54, 1996, pp. 1098-1105.
[17] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.
[18] H. A. Van Lint, Introduction to coding theory, (2nd Ed), Springer Verlag, (1992) pp. 34-35.
[19] J. Edminister, Schaum’s outline of theory and problems of electromagnetism, New York: McGraw-Hill Professional (1995) p. 225.
[2] A. Steane, “ Multiple-Particle Interference and Quantum Error Correction”, Proc. R. Soc. London, Ser. A, Vol. 452, No. 1954, 1996, pp. 2551-2577.
[3] P. Shor, “Scheme for reducing decoherence in a quantum computer memory”, Phys. Rev., Vol. A52, 1995, pp. 2493- 2496.
[4] D. Gottesman, “Class of quantum error-correcting codes saturating the quantum Hamming bound”, Phys. Rev., Vol. A54, 1996, pp. 1862-1868.
[5] A. R. Calderbank, E.M. Rains, P. W. Shor, and A. J. Sloane, “Quantum error correction and orthogonal geometry”, Phys. Rev., Vol. A78, 1997, pp. 405-409.
[6] A. R. Calderbank, E.M. Rains, P. W. Shor, and A. J. Sloane, “Quantum error correction via codes over gf(4)”, IEEE Trans. Inf. Theory, Vol. 44, 1998, pp. 1369-1372.
[7] P. Shor, “Fault-tolerant quantum computation”, Proceedings of the 37th Symposium on the Foundations of Computer Sciences (FOCS), 1995, 37 Los Alamitos, CA: IEEE Press, pp. 56-57.
[8] A. Y. Kitaev, “Quantum Error Corrections with Imperfect Gates”. In Quantum Communication and Computing and Measurement Edited by O. Hirota et al Plenum, New York.
[9] E. Knill, and R. Laflamme; “Concatenated Quantum Codes”, arxiv:quant-ph/9608012.
[10] D. Aharonov, and M. Ben-Or; “Fault-Tolerant Quantum Computation With Constant Error Rate”, arxiv:quantph/ 9906129.
[11] E. Knill, R. Laflamme, and W. H. Zurek, “Resilient Quantum Computation”, Science, Vol. 279, 1998, pp. 342-345.
[12] E. Knill, R. Laflamme, and W. H. Zurek, “Resilient quantum computation: error models and thresholds”, Proc. R. Soc. London Ser. A, Vol. 454, No. 1969, 1998, pp. 365-384.
[13] D. Gottesman, “Theory of fault-tolerant quantum computation”, Phys. Rev., Vol. A57, 1998, pp. 127-137.
[14] J. Preskill, “ Reliable quantum computers”, Proc. R. Soc. London, Ser. A, Vol. 454, No. 1969, 1998. pp. 385-410.
[15] A. Skander, M. Nadjim, and M. Benslama, “A New Error Correction Intuitive Approach in Quantum Communications Protocols”, Am. J. Appl. Sci., Vol. 4, No. 8, 2007, pp. 597-604.
[16] A. Calderbank, and P. Shor, “Good quantum errorcorrecting codes exist”, Phys. Rev. Vol. A 54, 1996, pp. 1098-1105.
[17] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.
[18] H. A. Van Lint, Introduction to coding theory, (2nd Ed), Springer Verlag, (1992) pp. 34-35.
[19] J. Edminister, Schaum’s outline of theory and problems of electromagnetism, New York: McGraw-Hill Professional (1995) p. 225.
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2009-10-01
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[1]
“Semiclassical grounds of the Calderbank-Shor-Steane quantum error correction codes”, JCS&T, vol. 9, no. 02, pp. p. 53–57, Oct. 2009, Accessed: Jan. 14, 2026. [Online]. Available: https://journal.info.unlp.edu.ar/JCST/article/view/717
