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| \tag*{(1)}\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, i.e., the regularized coherent information, is the highest rate of quantum information transmission through a quantum channel with arbitrarily small error rate [24]. See [5; Ch. 24] for definitions and a history.


See Refs. [610] for introductions to quantum error correction. See also tutorials by V. V. Albert, S. M. Girvin, P. Shor, and B. Terhal.Quantum error correction was initially claimed not to be theoretically possible [11,12].




  • Error-correcting code (ECC) — Error-correction conditions can similarly be formulated for classical codes [13; Sec. 3], although they are not as widely as used as those for quantum codes.
  • Metrological code — Metrological codes are logical-qubit codes that satisfy the Knill-Laflamme conditions conditions only partially, and codes that satisfy them fully are QECCs.
  • Subsystem quantum error-correcting code — A subsystem code reduces to an ordinary error-correcting code when the gauge subsystem is trivial, \(\mathsf{B} = \mathbb{C}\). Conversely, any QECC with a tensor-product logical subspace can be turned into a subsystem code by treating a logical tensor factor as a gauge subsystem.
  • \([[9,1,3]]\) Shor code — The Shor code is the first quantum error-correcting code.


C. Bény, A. Kempf, and D. W. Kribs, “Quantum error correction on infinite-dimensional Hilbert spaces”, Journal of Mathematical Physics 50, 062108 (2009) arXiv:0811.0421 DOI
S. Lloyd, “Capacity of the noisy quantum channel”, Physical Review A 55, 1613 (1997) arXiv:quant-ph/9604015 DOI
Peter W. Shor, The quantum channel capacity and coherent information, 2002 (obtained from the MSRI Workshop on Quantum Computation website).
I. Devetak, “The private classical capacity and quantum capacity of a quantum channel”, (2004) arXiv:quant-ph/0304127
“Preface to the Second Edition”, Quantum Information Theory xi (2016) arXiv:1106.1445 DOI
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) arXiv:1302.3428 DOI
J. Roffe, “Quantum error correction: an introductory guide”, Contemporary Physics 60, 226 (2019) arXiv:1907.11157 DOI
W. G. Unruh, “Maintaining coherence in quantum computers”, Physical Review A 51, 992 (1995) arXiv:hep-th/9406058 DOI
“Is quantum mechanics useful?”, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences 353, 367 (1995) DOI
B. Yoshida, “Decoding the Entanglement Structure of Monitored Quantum Circuits”, (2021) arXiv:2109.08691
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: qecc

Cite as:
“Quantum error-correcting code (QECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qecc
  title={Quantum error-correcting code (QECC)},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:

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.