Error Correcting Codes in Quantum Theory

A. M. Steane

doi:10.1103/PhysRevLett.77.793

Featured in Physics
Access by Xinjiang University

Error Correcting Codes in Quantum Theory

A. M. Steane

Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, England

PDF

Phys. Rev. Lett. 77, 793 – Published 29 July, 1996

DOI: https://doi.org/10.1103/PhysRevLett.77.793

Abstract

A new type of uncertainty relation is presented, concerning the information-bearing properties of a discrete quantum system. A natural link is then revealed between basic quantum theory and the linear error correcting codes of classical information theory. A subset of the known codes is described, having properties which are important for error correction in quantum communication. It is shown that a pair of states which are, in a certain sense, “macroscopically different,” can form a superposition in which the interference phase between the two parts is measurable. This provides a highly stabilized “Schrödinger cat” state.

Focus

Landmarks—Correcting Quantum Computer Errors

Published 10 June, 2016

In the mid-1990s, researchers proposed methods to preserve the integrity of quantum bits—techniques that may become the key to practical quantum computing on a large scale.

See more in Physics

Special Feature

Quantum Milestones, 1995: Correcting Quantum Computer Errors

Published 14 March, 2025

Researchers proposed methods to preserve the integrity of quantum bits—techniques that may become the key to practical quantum computing on a large scale.

See more in Physics

References (15)

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam,1977).
For reviews and references, see, e.g., A. Ekert, in A. Ekert and R. JozsaAtomic Physics 14, D. J. Wineland, C. E. Wieman, and S. J. Smith (AIP Press, New York, 1995); (to be published).
For reviews, see, e.g., R. J. Hughes, D. M. Alde, P. Dyer, G. G. Luther, G. L. Morgan, and M. Schauer, Contemp. Phys. 36, 149 (1995) S. J. D. Phoenix and P. D. Townsend, 36, 165 (1995).
E. Schrödinger, Naturwissenschaften 23, 807 (1935); translated in Quantum Theory and Measurement, J. A. Wheeler and W. H. Zurek (Princeton University, Princeton, NJ, 1983).
D. Deutsch, Proc. R. Soc. London A 400, 97 (1985).
I. Bialynicki and J. Mycielski, Commun. Math. Phys. 44, 129 (1975).
D. Deutsch, Phys. Rev. Lett. 50, 631 (1983).
H. Maassen and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988).
R. W. Hamming, Bell Syst. Tech. J 29, 147 (1950).
N. David Mermin, Phys. Rev. Lett. 65, 1838 (1990).
A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, Dordrecht,1993).
More precisely, the probability that $φ$ is revealed in the correct one of the two ranges $(- π / 2, \dots, π / 2), (π / 2, \dots, 3 π / 2)$ in a given experimental run is $P = Σ {i = 0}^{⌊ n / 2 ⌋} ({n} {2 i}) p^{2 i} (1 - p)^{n - 2 i}$ (the probability that zero or an even number of errors occur), which approaches $1 / 2$ very rapidly as $n$ increases. For example, if $p = 0.02, n = 1001$ then $| P - 1 / 2 | \sim 10^{- 14}$ . When $P = 1 / 2$ the experimental results bear no correlation to $φ$ .
A. R. Calderbank and P. W. Shor, Phys. Rev. A (to be published).
A. M. Steane, Proc. R. Soc. London A (to be published).
P. W. Shor, Phys. Rev. A 52, R2493 (1995).

[c1] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam,1977).

[c2] For reviews and references, see, e.g., A. Ekert, in A. Ekert and R. JozsaAtomic Physics 14, D. J. Wineland, C. E. Wieman, and S. J. Smith (AIP Press, New York, 1995); (to be published).

[c3] For reviews, see, e.g., R. J. Hughes, D. M. Alde, P. Dyer, G. G. Luther, G. L. Morgan, and M. Schauer, Contemp. Phys. 36, 149 (1995) S. J. D. Phoenix and P. D. Townsend, 36, 165 (1995).

[c4] E. Schrödinger, Naturwissenschaften 23, 807 (1935); translated in Quantum Theory and Measurement, J. A. Wheeler and W. H. Zurek (Princeton University, Princeton, NJ, 1983).

[c5] D. Deutsch, Proc. R. Soc. London A 400, 97 (1985).

[c6] I. Bialynicki and J. Mycielski, Commun. Math. Phys. 44, 129 (1975).

[c7] D. Deutsch, Phys. Rev. Lett. 50, 631 (1983).

[c8] H. Maassen and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988).

[c9] R. W. Hamming, Bell Syst. Tech. J 29, 147 (1950).

[c10] N. David Mermin, Phys. Rev. Lett. 65, 1838 (1990).

[c11] A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, Dordrecht,1993).

[c12] More precisely, the probability that $φ$ is revealed in the correct one of the two ranges $(- π / 2, \dots, π / 2), (π / 2, \dots, 3 π / 2)$ in a given experimental run is $P = Σ {i = 0}^{⌊ n / 2 ⌋} ({n} {2 i}) p^{2 i} (1 - p)^{n - 2 i}$ (the probability that zero or an even number of errors occur), which approaches $1 / 2$ very rapidly as $n$ increases. For example, if $p = 0.02, n = 1001$ then $| P - 1 / 2 | \sim 10^{- 14}$ . When $P = 1 / 2$ the experimental results bear no correlation to $φ$ .

[c13] A. R. Calderbank and P. W. Shor, Phys. Rev. A (to be published).

[c14] A. M. Steane, Proc. R. Soc. London A (to be published).

[c15] P. W. Shor, Phys. Rev. A 52, R2493 (1995).

Physical Review Letters

Export citation

Export citation

Error Correcting Codes in Quantum Theory

Abstract

Focus

Landmarks—Correcting Quantum Computer Errors

Special Feature

Quantum Milestones, 1995: Correcting Quantum Computer Errors

References (15)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation