## Description

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.

Since quantum information is encoded in quantum superpositions, an additional source of noise (not relevant to classical encodings) can affect the relative phase of such superpositions. Quantum error-correcting codes have to protect against such phase-flip noise while also protecting against conventional classical bit-flip noise. Codes are not expected to protect against both types of noise perfectly, and there is generally a tradeoff.

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.

## Protection

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].

## Rate

## Notes

## Parents

## Children

- Group-based quantum code
- Approximate quantum error-correcting code (AQECC)
- Asymmetric quantum code
- Block quantum code
- Group-representation code — Group-representation code projections are onto an irrep of a subgroup of canonical or distinguished unitary operations on a Hilbert space. Removing the restriction to distinguished operations and allowing all operations, every code projection on an \(N\)-dim Hilbert space can be expressed as a projection onto the irrep formed by the code-preserving subgroup of \(U(N)\). The same idea holds when \(N\) is taken to infinity.
- Hamiltonian-based code
- Finite-dimensional quantum error-correcting code
- Concatenated quantum code
- Random quantum code
- Monolithic quantum code

## Cousins

- Error-correcting code (ECC) — Quantum information cannot be copied using a linear process [21], so one cannot send several copies of a quantum state through a channel as can be done for classical information. Error-correction conditions can similarly be formulated for classical codes [22; 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.
- Entanglement-assisted (EA) QECC — EA QECCs utilize additional ancillary subsystems in a pre-shared entangled state, but reduce to QECCs when said subsystems are interpreted as noiseless physical subsystems.
- Subsystem QECC — 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.
- Error-transmuting code (QETC) — QETCs are quantum codes which satisfy a generalization of the Knill-Laflamme conditions. QETCs for which the admissible logical error set consists solely of the identity are QECCs.
- \([[9,1,3]]\) Shor code — The Shor code is the first quantum error-correcting code.

## References

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- C. Bény, A. Kempf, and D. W. Kribs, “Quantum error correction on infinite-dimensional Hilbert spaces”, Journal of Mathematical Physics 50, (2009) arXiv:0811.0421 DOI
- [2]
- S. Lloyd, “Capacity of the noisy quantum channel”, Physical Review A 55, 1613 (1997) arXiv:quant-ph/9604015 DOI
- [3]
- Peter W. Shor, The quantum channel capacity and coherent information, 2002 (obtained from the MSRI Workshop on Quantum Computation website).
- [4]
- I. Devetak, “The private classical capacity and quantum capacity of a quantum channel”, (2004) arXiv:quant-ph/0304127
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- “Preface to the Second Edition”, Quantum Information Theory xi (2016) arXiv:1106.1445 DOI
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- A. Kitaev, A. Shen, and M. Vyalyi, Classical and Quantum Computation (American Mathematical Society, 2002) DOI
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- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2012) DOI
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- S. J. Devitt, W. J. Munro, and K. Nemoto, “Quantum error correction for beginners”, Reports on Progress in Physics 76, 076001 (2013) arXiv:0905.2794 DOI
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- B. M. Terhal, “Quantum error correction for quantum memories”, Reviews of Modern Physics 87, 307 (2015) arXiv:1302.3428 DOI
- [10]
- D. Suter and G. A. Álvarez, “Colloquium: Protecting quantum information against environmental noise”, Reviews of Modern Physics 88, (2016) DOI
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- M. Grassl, “Classical Information Theory and Classical Error Correction”, Quantum Information 1 (2016) DOI
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- J. Roffe, “Quantum error correction: an introductory guide”, Contemporary Physics 60, 226 (2019) arXiv:1907.11157 DOI
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- J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
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- T. A. Brun, “Quantum Error Correction”, (2019) arXiv:1910.03672
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- “Quantum Error Correction”, Artificial Intelligence and Quantum Computing for Advanced Wireless Networks 451 (2022) DOI
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- D. A. Lidar and T. A. Brun, editors , Quantum Error Correction (Cambridge University Press, 2013) DOI
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- F. Gaitan, Quantum Error Correction and Fault Tolerant Quantum Computing (CRC Press, 2018) DOI
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- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
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- W. G. Unruh, “Maintaining coherence in quantum computers”, Physical Review A 51, 992 (1995) arXiv:hep-th/9406058 DOI
- [20]
- “Is quantum mechanics useful?”, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences 353, 367 (1995) DOI
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- W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned”, Nature 299, 802 (1982) DOI
- [22]
- B. Yoshida, “Decoding the Entanglement Structure of Monitored Quantum Circuits”, (2021) arXiv:2109.08691

## Page edit log

- Victor V. Albert (2022-07-30) — most recent
- Philippe Faist (2022-07-15)
- Victor V. Albert (2022-01-03)

## 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/edit/main/codes/quantum/properties/qecc.yml.