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. The state space of a QECC is contained in the space of complex \(L^2\)-normalizable functions of some configuration space, which usually corresponds to the alphabet of a classical code.
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.
While all considered QECC states are complex functions, real or quaternionic function spaces can also be considered for QEC [1,2].
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 [3].
Pseudo-threshold (a.k.a. break-even point): The ultimate goal of error correction is to make sure that the logical error rate is greater than the underlying physical error rate. For a noise model parameterized by a single physical error rate \(p\), the pseudo-threshold or break-even point is the smallest \(p\) at which the logical error rate after error correction is equal to \(p\).
Rate
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 [4–6]. In other words, the capacity formula implies that one can achieve a transmission rate \(r\) over a quantum channel \(\mathcal{E}\) iff, for sufficiently large \(n\), \(m=\lfloor r n \rfloor\), and for all \(\epsilon>0\), \begin{align} ||\mathcal{D}\mathcal{E}\mathcal{U}-I^{\otimes m}||_1\leq \epsilon \tag*{(2)}\end{align} for some encoding channel \(\mathcal{U}\) and some recovery channel \(\mathcal{D}\). The quantum capacity \(Q\) of \(\mathcal{E}\) is defined as the supremum over \(n\) of achievable transmission rates [7]. See [8; Ch. 24] for definitions and a history.
The fault-tolerant capacity is the capacity for the more general case where the encoding and decoding maps are also assumed to undergo noise [9].
Doublin coefficients [10] for quantum channels have been studied [11].
Decoding
The effect of an error is a mapping of the code subspace into another, potentially overlapping, subspace. To determine, or diagnose, the effect of the error in what is known as syndrome-based decoding, one can measure one or more operators called check operators, which resolve code and error spaces without collapsing the quantum information inside the spaces. The eigenvalues of check operators are called error syndromes. One round or cycle of quantum error correction proceeds by extracting syndromes and performing correcting operations to map the error space containing the logical information back into the codespace. For some codes, correcting operations are not necessary because one can instead track which error space contains the logical information.Notes
See Refs. [12–28] for overviews of quantum error correction.See Refs. [29–31] for books on quantum error correction.See video tutorials by V. V. Albert, S. M. Girvin, P. Shor, B. Terhal, and J. Wright.Quantum error correction was initially claimed not to be theoretically possible [32,33] and has been criticized since [34].Resource-theoretic interpretations of quantum error correction have been developed, including those that think of codes together with recovery operations as superchannels (a.k.a. quantum combs or bipartite operations) [35–39].Cousins
- Approximate quantum error-correcting code (AQECC)— QAECCs correcting a noise channel exactly reduce to QECCs.
- Error-correcting code (ECC)— Quantum information cannot be copied using a linear process [40], so one cannot send several copies of a quantum state through a channel as can be done for classical information. The Knill-Laflamme conditions can similarly be formulated for classical codes [41; 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.
- Hybrid QECC— A hybrid QECC storing no classical information reduces to a QECC. Conversely, any QECC can be converted into a hybrid QECC by using a portion of its logical subspace to store only classical information.
- Subsystem QECC— A subsystem QECC reduces to an ordinary (i.e., subspace) QECC when the gauge subsystem is trivial. 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.
- Holographic tensor-network code— Quantum encoding maps are isometries, but non-isometric encodings are relevant to describing mappings into the interior of a black hole [42] and de Sitter time evolution [43]. Trace-norm preserving encodings have also been studied [44].
- Quantum 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.
Primary Hierarchy
References
- [1]
- A. Yu. Vlasov, “Error Correction with Euclidean Qubits”, (1999) arXiv:quant-ph/9911074
- [2]
- V. Nyirahafashimana, N. M. Shah, U. A. Halim, M. Othman, and S. K. S. Husain, “Quantum Error Correction in Quaternionic Hilbert Spaces”, (2025) arXiv:2504.19833
- [3]
- 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
- [4]
- S. Lloyd, “Capacity of the noisy quantum channel”, Physical Review A 55, 1613 (1997) arXiv:quant-ph/9604015 DOI
- [5]
- Peter W. Shor, The quantum channel capacity and coherent information, 2002 (obtained from the MSRI Workshop on Quantum Computation website).
- [6]
- I. Devetak, “The private classical capacity and quantum capacity of a quantum channel”, (2004) arXiv:quant-ph/0304127
- [7]
- J. Watrous, The Theory of Quantum Information (Cambridge University Press, 2018) DOI
- [8]
- “Preface to the Second Edition”, Quantum Information Theory xi (2016) arXiv:1106.1445 DOI
- [9]
- M. Christandl and A. Müller-Hermes, “Fault-Tolerant Coding for Quantum Communication”, IEEE Transactions on Information Theory 70, 282 (2024) arXiv:2009.07161 DOI
- [10]
- Doeblin, W. (1937). Sur les propriétés asymptotiques de mouvement régis par certains types de chaines simples. Bulletin mathématique de la Société roumaine des sciences, 39(1), 57-115.
- [11]
- A. Makur and J. Singh, “Doeblin Coefficients and Related Measures”, IEEE Transactions on Information Theory 70, 4667 (2024) arXiv:2309.08475 DOI
- [12]
- J. Preskill, “Fault-tolerant quantum computation”, (1997) arXiv:quant-ph/9712048
- [13]
- D. Gottesman, “An Introduction to Quantum Error Correction”, (2000) arXiv:quant-ph/0004072
- [14]
- A. Kitaev, A. Shen, and M. Vyalyi, Classical and Quantum Computation (American Mathematical Society, 2002) DOI
- [15]
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2012) DOI
- [16]
- D. Gottesman, “Quantum Error Correction and Fault-Tolerance”, (2005) arXiv:quant-ph/0507174
- [17]
- J. Kempe, “Approaches to Quantum Error Correction”, (2006) arXiv:quant-ph/0612185
- [18]
- D. Gottesman, “An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation”, (2009) arXiv:0904.2557
- [19]
- 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
- [20]
- B. M. Terhal, “Quantum error correction for quantum memories”, Reviews of Modern Physics 87, 307 (2015) arXiv:1302.3428 DOI
- [21]
- D. Suter and G. A. Álvarez, “Colloquium: Protecting quantum information against environmental noise”, Reviews of Modern Physics 88, (2016) DOI
- [22]
- M. Grassl, “Classical Information Theory and Classical Error Correction”, Quantum Information 1 (2016) DOI
- [23]
- A. Paler and S. J. Devitt, “An introduction to Fault-tolerant Quantum Computing”, (2015) arXiv:1508.03695
- [24]
- J. Roffe, “Quantum error correction: an introductory guide”, Contemporary Physics 60, 226 (2019) arXiv:1907.11157 DOI
- [25]
- J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
- [26]
- T. A. Brun, “Quantum Error Correction”, Oxford Research Encyclopedia of Physics (2020) arXiv:1910.03672 DOI
- [27]
- “Quantum Error Correction”, Artificial Intelligence and Quantum Computing for Advanced Wireless Networks 451 (2022) DOI
- [28]
- P. J. Nadkarni, N. Rengaswamy, and B. Vasić, “Tutorial on Quantum Error Correction for 2024 Quantum Information Knowledge (QuIK) Workshop”, (2024) arXiv:2407.12737
- [29]
- D. A. Lidar and T. A. Brun, editors , Quantum Error Correction (Cambridge University Press, 2013) DOI
- [30]
- F. Gaitan, Quantum Error Correction and Fault Tolerant Quantum Computing (CRC Press, 2018) DOI
- [31]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [32]
- W. G. Unruh, “Maintaining coherence in quantum computers”, Physical Review A 51, 992 (1995) arXiv:hep-th/9406058 DOI
- [33]
- “Is quantum mechanics useful?”, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences 353, 367 (1995) DOI
- [34]
- R. Alicki, “Critique of Fault-Tolerant Quantum Information Processing”, (2013) arXiv:1310.8457
- [35]
- O. Oreshkov, F. Costa, and Č. Brukner, “Quantum correlations with no causal order”, Nature Communications 3, (2012) arXiv:1105.4464 DOI
- [36]
- W. Matthews and S. Wehner, “Finite Blocklength Converse Bounds for Quantum Channels”, IEEE Transactions on Information Theory 60, 7317 (2014) arXiv:1210.4722 DOI
- [37]
- D. Leung and W. Matthews, “On the Power of PPT-Preserving and Non-Signalling Codes”, IEEE Transactions on Information Theory 61, 4486 (2015) arXiv:1406.7142 DOI
- [38]
- A. Tanggara, M. Gu, and K. Bharti, “Strategic Code: A Unified Spatio-Temporal Framework for Quantum Error-Correction”, (2024) arXiv:2405.17567
- [39]
- D.-S. Wang, Y.-D. Liu, Y.-J. Wang, and S. Luo, “Quantum resource theory of coding for error correction”, Physical Review A 110, (2024) arXiv:2409.09416 DOI
- [40]
- W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned”, Nature 299, 802 (1982) DOI
- [41]
- B. Yoshida, “Decoding the Entanglement Structure of Monitored Quantum Circuits”, (2021) arXiv:2109.08691
- [42]
- C. Akers, N. Engelhardt, D. Harlow, G. Penington, and S. Vardhan, “The black hole interior from non-isometric codes and complexity”, (2022) arXiv:2207.06536
- [43]
- J. Cotler and A. Strominger, “The Universe as a Quantum Encoder”, (2022) arXiv:2201.11658
- [44]
- F. Ticozzi and L. Viola, “Quantum information encoding, protection, and correction from trace-norm isometries”, Physical Review A 81, (2010) arXiv:0912.0963 DOI
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.