## Good quantum error-correcting codes exist

Publication
, Journal Article

Calderbank, AR; Shor, PW

Published in: Physical Review A - Atomic, Molecular, and Optical Physics

January 1, 1996

A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (two-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits. Quantum error-correcting codes are shown to exist with asymptotic rate k/n=1-2[Formula Presented](2t/n) where [Formula Presented](p) is the binary entropy function -p[Formula Presented]p-(1-p)[Formula Presented](1-p). Upper bounds on this asymptotic rate are given. © 1996 The American Physical Society.

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## Published In

Physical Review A - Atomic, Molecular, and Optical Physics

## DOI

## EISSN

1094-1622

## ISSN

1050-2947

## Publication Date

January 1, 1996

## Volume

54

## Issue

2

## Start / End Page

1098 / 1105

## Related Subject Headings

- General Physics
- 03 Chemical Sciences
- 02 Physical Sciences
- 01 Mathematical Sciences

### Citation

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Calderbank, A. R., & Shor, P. W. (1996). Good quantum error-correcting codes exist.

*Physical Review A - Atomic, Molecular, and Optical Physics*,*54*(2), 1098–1105. https://doi.org/10.1103/PhysRevA.54.1098Calderbank, A. R., and P. W. Shor. “Good quantum error-correcting codes exist.”

*Physical Review A - Atomic, Molecular, and Optical Physics*54, no. 2 (January 1, 1996): 1098–1105. https://doi.org/10.1103/PhysRevA.54.1098.Calderbank AR, Shor PW. Good quantum error-correcting codes exist. Physical Review A - Atomic, Molecular, and Optical Physics. 1996 Jan 1;54(2):1098–105.

Calderbank, A. R., and P. W. Shor. “Good quantum error-correcting codes exist.”

*Physical Review A - Atomic, Molecular, and Optical Physics*, vol. 54, no. 2, Jan. 1996, pp. 1098–105.*Scopus*, doi:10.1103/PhysRevA.54.1098.Calderbank AR, Shor PW. Good quantum error-correcting codes exist. Physical Review A - Atomic, Molecular, and Optical Physics. 1996 Jan 1;54(2):1098–1105.

## Published In

Physical Review A - Atomic, Molecular, and Optical Physics

## DOI

## EISSN

1094-1622

## ISSN

1050-2947

## Publication Date

January 1, 1996

## Volume

54

## Issue

2

## Start / End Page

1098 / 1105

## Related Subject Headings

- General Physics
- 03 Chemical Sciences
- 02 Physical Sciences
- 01 Mathematical Sciences