×

zbMATH — the first resource for mathematics

Theory of quantum error correction for general noise. (English) Zbl 0956.81008
Summary: A measure of quality of an error-correcting code is the maximum number of errors that it is able to correct. We show that a suitable notion of “number of errors” \(e\) makes sense for any quantum or classical system in the presence of arbitrary interactions. Thus, \(e\)-error-correcting codes protect information without requiring the usual assumptions of independence. We prove the existence of large codes for both quantum and classical information. By viewing error-correcting codes as subsystems, we relate codes to irreducible representations of operator algebras and show that noiseless subsystems are infinite-distance error-correcting codes.

MSC:
81P68 Quantum computation
94B60 Other types of codes
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] P. W. Shor, in: Proceedings of the Symposium on the Foundations of Computer Science (1996)
[2] D. Aharonov, in: Proceedings of the 29th Annual ACM Symposium on the Theory of Computing (1996)
[3] A. Yu. Kitaev, Usp. Mat. Nauk. 52 pp 53– (1997) · doi:10.4213/rm892
[4] A. Yu. Kitaev, Russ. Math. Survey 52 pp 1191– (1997) · Zbl 0917.68063 · doi:10.1070/RM1997v052n06ABEH002155
[5] E. Knill, Science 279 pp 342– (1998) · doi:10.1126/science.279.5349.342
[6] P. W. Shor, Phys. Rev. A 52 pp R2493– (1995) · doi:10.1103/PhysRevA.52.R2493
[7] A. Steane, Proc. R. Soc. London A 452 pp 2551– (1996) · Zbl 0876.94002 · doi:10.1098/rspa.1996.0136
[8] E. Knill, Phys. Rev. A 55 pp 900– (1997) · doi:10.1103/PhysRevA.55.900
[9] P. Zanardi, Phys. Rev. Lett. 79 pp 3306– (1997) · doi:10.1103/PhysRevLett.79.3306
[10] D. A. Lidar, Phys. Rev. Lett. 81 pp 2594– (1998) · doi:10.1103/PhysRevLett.81.2594
[11] W. Magnus, in: Combinatorial Group Theory (1976)
[12] C. H. Bennett, Phys. Rev. A 54 pp 3824– (1996) · Zbl 1371.81041 · doi:10.1103/PhysRevA.54.3824
[13] M. A. Nielsen, Proc. R. Soc. London A 454 pp 277– (1998) · Zbl 0982.94006 · doi:10.1098/rspa.1998.0160
[14] H. Tverberg, J. London Math. Soc. 41 pp 123– (1966) · Zbl 0131.20002 · doi:10.1112/jlms/s1-41.1.123
[15] C. W. Curtis, in: Representation Theory of Finite Groups and Associative Algebras (1962) · doi:10.1090/chel/356
[16] B. Schumacher, Phys. Rev. A 54 pp 2614– (1996) · doi:10.1103/PhysRevA.54.2614
[17] R. Alicki, in: Quantum Dynamical Semigroups and Applications (1987) · Zbl 0652.46055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.