Quantum error-correcting code (QECC)


Encodes quantum information in a (logical) subspace of a (physical) Hilbert space such that it is possible to recover said information from errors that act as linear maps on the physical space. The logical subspace is spanned by a basis comprised of code basis states or codewords. Codewords may not be normalizable if the physical Hilbert space in infinite, so approximate versions have to be constructed in practice.


Denoting Hilbert spaces by the letter \(\mathsf{H}\), a quantum code \((U,\cal{E})\) is a partial isometry \(U:\mathsf{H}_{\text{logical}}\to\mathsf{H}_{\text{physical}}\) with a set of correctable errors \(\cal{E}\) with the following property: there exists a quantum operation \(\cal{D}\) such that for all \(E\in\cal{E}\) and states \(|\psi\rangle\in\mathsf{H}_{\text{logical}}\), \begin{align} {\cal D} (EU|\psi\rangle\langle\psi|U^{\dagger}E^{\dagger}) = c(E,|\psi\rangle)|\psi\rangle\langle\psi| \end{align} for some constant \(c\).

Equivalently, correction capability is determined by the Knill-Laflamme conditions, which may admit infinite terms due to non-normalizability of ideal code states in the case of codes with infinite-dimenional physical spaces. A code that satisfies these conditions approximately, i.e., up to some small quantifiable error, is called an approximate code. These conditions can also be formulated in terms of a dual Heisenberg picture, where correctability is checked for some algebra of observables [1].


The quantum channel capacity is the highest rate of quantum information transmission through a quantum channel with arbitrarily small error rate; see [2; Ch. 24] for definitions and a history.


See Refs. [3][4][5][6][7] for introductions to quantum error correction.





C. Bény, A. Kempf, and D. W. Kribs, “Quantum error correction on infinite-dimensional Hilbert spaces”, Journal of Mathematical Physics 50, 062108 (2009). DOI; 0811.0421
“Preface to the Second Edition”, Quantum Information Theory xi (2016). DOI; 1106.1445
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2012). DOI
J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
D. Gottesman. Surviving as a quantum computer in a classical world
B. M. Terhal, “Quantum error correction for quantum memories”, Reviews of Modern Physics 87, 307 (2015). DOI; 1302.3428
J. Roffe, “Quantum error correction: an introductory guide”, Contemporary Physics 60, 226 (2019). DOI; 1907.11157
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“Quantum error-correcting code (QECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qecc
@incollection{eczoo_qecc, title={Quantum error-correcting code (QECC)}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qecc} }
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Cite as:

“Quantum error-correcting code (QECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qecc

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/properties/qecc.yml.