Description
Encodes quantum information in a \(K\)-dimensional (logical) subspace of an \(N\)-dimensional (physical) Hilbert space such that it is possible to recover said information from errors. The logical subspace is spanned by a basis comprised of code basis states or codewords.Protection
Denoting Hilbert spaces by the letter \(\mathsf{H}\), a finite-dimensional quantum code \((U,\cal{E})\) is a partial isometry \(U:\mathsf{H}_{K}\to\mathsf{H}_{N}\) and a set of correctable errors \({\cal{E}}:\mathsf{H}_N\to\mathsf{H}_M\) with the following property: there exists a quantum operation \({\cal{D}}:\mathsf{H}_M\to \mathsf{H}_K\) such that for all \(E\in\cal{E}\) and states \(|\psi\rangle\in\mathsf{H}_{K}\), \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 \(c\) [1]. A code is said to protect against or correct the errors \(\mathcal{E}\).
Knill-Laflamme error-correction conditions
Equivalently, correction capability is determined by the quantum error-correction conditions. A code that satisfies these conditions approximately, i.e., up to some small quantifiable error, is called an approximate code.
Knill-Laflamme conditions: The Knill-Laflamme error-correction conditions [2–4][5; Thm. 10.1] are necessary and sufficient conditions for a code to successfully correct a set of errors in a finite-dimensional Hilbert space. A code (defined by a partial isometry \(U\)) with code space projector \(\Pi = U U^\dagger\) can correct a set of errors \(\{ E_j \}\) if and only if \begin{align} \Pi E_i^\dagger E_j \Pi = c_{ij}\, \Pi\qquad\text{for all \(i,j\),} \tag*{(2)}\end{align} where the QEC matrix elements \(c_{ij}\) are arbitrary complex numbers. The term \(\Pi E_i^\dagger E_j \Pi\) can be split into two types of conditions, the diagonal (a.k.a. non-deformation or invariance) conditions \(\langle \psi| E_i^\dagger E_j | \psi\rangle\) for a codeword \(|\psi\rangle\), and the off-diagonal (a.k.a. orthogonality or distinguishability) conditions \(\langle \psi| E_i^\dagger E_j | \phi\rangle\) for two orthogonal codewords \(|\psi\rangle\) and \(|\phi\rangle\). By linearity of quantum error correction, if a code corrects a set of errors \(\mathcal{E}\), then it also corrects \(\operatorname{span}\mathcal{E}\).
For codewords whose basis states are chosen far enough apart (in some notion of distance) so that the off-diagonal conditions are automatically zero, the remaining diagonal conditions correspond to a system of non-linear constraints on the basis-expansion coefficients of the codewords. In this setting, there exist finite QECCs with \(K = \lceil N/D\rceil/(D+1)\) that protect against an error set with \(D\) basis elements [6; Thm. 4], a consequence of the Tverberg theorem [7–9]. For \(K = 2\) this theorem reduces to Radon’s theorem [8,10].
The Knill-Laflamme conditions can alternatively be expressed in terms of the complementary channel, in an entropic information-theoretic way via a data processing inequality [11–15], or can be interpreted thermodynamically [16]. They motivate higher-rank numerical ranges, which are generalizations of the numerical range of an operator [17–19]. They have been extended to sequences of multiple errors and rounds of correction [20].
Degeneracy: A code is degenerate with respect to a noise model if different errors map code states to the same error subspace. For a linearly independent error set \(\cal{E}\), degeneracy is equivalent to \(\text{rank}(c_{ij}) < |\cal{E}|\) [1].
Correctability of quantum channels
From now on, we use \(\mathcal{E}\) to denote a noise channel constructed out of the set of errors \(E\) and let \(\mathcal{U}(\cdot)=U(\cdot)U^\dagger\) be the superoperator corresponding to the partial encoding isometry \(U\). A noise channel is correctable if there exists a recovery channel \(\mathcal{D}\) such that \begin{align} \mathcal{D}\mathcal{E}\mathcal{U}(\rho)=\rho \tag*{(3)}\end{align} for all logical states \(\rho\).
The above is equivalent to the fidelity between \(\rho\) and \(\mathcal{D}\mathcal{E}\mathcal{U}(\rho)\) being one for any notion of distance between quantum states. In particular, we can consider a scenario where we send only one part of an entangled state through a channel and determine whether the entanglement has been preserved during transmission. Using the notion of entanglement fidelity, a quantum channel \(\mathcal{E}\) is exactly correctable iff there exists a quantum channel \(\mathcal{D}\) such that \begin{align} (\mathcal{D}\mathcal{E}\mathcal{U}\otimes\mathrm{id})(\ket{\psi}\bra{\psi})=\ket{\psi}\bra{\psi} \tag*{(4)}\end{align} for all states \(\rho\) and their corresponding purifications \(\ket{\psi}\) (i.e., states \(\ket{\psi}\) such that \(\text{Tr}_{2}(|\psi\rangle\langle\psi|)=\rho\)).
The above entanglement fidelity condition can be alternatively expressed using complementary channels.
Complementary channel: A complementary channel \(\mathcal{E}^C\) is obtained from a channel \(\mathcal{E}\) that acts on a system by interpreting the channel as coming from a unitary operation acting on a larger system-environment tensor-product space (i.e., performing an isometric extension) with the environment necessarily in a pure state, and then tracing out the system factor (instead of the second environmental factor) [21; Sec. 5.2.2]. A noise channel \({\cal E}(\cdot)=\sum_{j}E_{j}(\cdot)E_{j}^{\dagger}\) admits a complementary channel of the form \begin{align} {\cal E}^{C}(\cdot)=\sum_{j,k}\text{Tr}\{E_{j}(\cdot)E_{k}^{\dagger}\}|j\rangle\langle k|~. \tag*{(5)}\end{align}
A channel \(\mathcal{E}\) is correctable if \(\mathcal{E}^C(\rho)=\rho_0\mathrm{Tr}(\rho)\) for some constant state \(\rho_0\), which is equivalent to the Knill-Laflamme conditions [22,23]. The logical and physical dimensions are related to the channel rank for non-degenerate codes via the quantum packing bound [24].
Exact correctability can also be expressed using the coherent information.
Coherent information: Given a bipartite state \(\rho_{RQ}\), the coherent information in subsystem \(Q\) is \begin{align} I_{c}(\rho_{RQ})=S(\rho_{Q})-S(\rho_{RQ})~. \tag*{(6)}\end{align} For a channel \(\mathcal{E}:L\to Q\) and a pure input state \(\rho\) on \(R\otimes L\), the coherent information of the channel is \begin{align} I_{c}(\mathcal{E},\rho)=S(\mathcal{E}(\rho_{L}))-S((\mathrm{id}\otimes\mathcal{E})(\rho))~. \tag*{(7)}\end{align} Coherent information cannot increase under further processing of the output, a statement known as the quantum data processing inequality [1,11–15].
Exact correctability is equivalent to preservation of coherent information: a channel \(\mathcal{E}\) is exactly correctable on a code iff the coherent information after encoding and noise is the same as that of the logical input for every pure input state, and it is enough to check this on a maximally entangled state between the logical system and a reference [1,11,13].
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 [25–27]. 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} \lVert\mathcal{D}\mathcal{E}\mathcal{U}-I^{\otimes m}\rVert_1\leq \epsilon \tag*{(8)}\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 [28]. See [29; 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 [30].
Doeblin coefficients [31] for quantum channels have been studied [32].
Decoding
The operation \(\cal{D}\) in the definition of this code is called the decoder. However, the term decoder can sometimes be used for the unencoder \(\cal{U}\) (i.e., the inverse of the encoder), which does not correct errors.There are several recovery maps which work for noise that is not exactly correctable; see AQECC entry.QECCs are useful [33] for the mean king’s measurement problem [34].Protection can be implemented via autonomous QEC (a.k.a. continuous QEC or continuous-time QEC) [35–39] via, e.g., reservoir engineering [40]; see review [41]. There are analogues of the Knill-Laflamme conditions for autonomous QEC [42,43], and it has been adapted to non-Markovian noise [44]. Information-theoretic bounds have been derived for open-loop control [45]. Machine learning can be used to optimize autonomous QEC encoding and recovery [46].Code Capacity Threshold
Coherent information of the state under the action of a noise channel can be used to estimate the optimal threshold [47].Cousins
- Finite-dimensional error-correcting code (ECC)— Finite-dimensional QECCs are quantum analogues of finite-dimensional classical ECCs.
- Complex projective space code— Pure quantum states in an \((N+1)\)-dimensional Hilbert space are parameterized by points in the complex projective space \(\mathbb{C}P^N\). As such, (classical) complex projective codes can be associated with subsets of pure quantum states.
Primary Hierarchy
References
- [1]
- D. Gottesman, Surviving as a Quantum Computer in a Classical World (2024) URL
- [2]
- E. Knill and R. Laflamme, “Theory of quantum error-correcting codes”, Physical Review A 55, 900 (1997) DOI
- [3]
- C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996) arXiv:quant-ph/9604024 DOI
- [4]
- J. Preskill, Lecture notes on Quantum Computation (1997–2020) URL
- [5]
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2012) DOI
- [6]
- E. Knill, R. Laflamme, and L. Viola, “Theory of Quantum Error Correction for General Noise”, (1999) arXiv:quant-ph/9908066
- [7]
- H. Tverberg, “A Generalization of Radon’s Theorem”, Journal of the London Mathematical Society s1-41, 123 (1966) DOI
- [8]
- J. Matoušek, editor , Lectures on Discrete Geometry (Springer New York, 2002) DOI
- [9]
- I. Bárány and P. Soberón, “Tverberg’s theorem is 50 years old: a survey”, (2018) arXiv:1712.06119
- [10]
- J. Radon, “Mengen konvexer K�rper, die einen gemeinsamen Punkt enthalten”, Mathematische Annalen 83, 113 (1921) DOI
- [11]
- B. Schumacher and M. A. Nielsen, “Quantum data processing and error correction”, Physical Review A 54, 2629 (1996) arXiv:quant-ph/9604022 DOI
- [12]
- E. Knill, R. Laflamme, and L. Viola, “Theory of Quantum Error Correction for General Noise”, Physical Review Letters 84, 2525 (2000) arXiv:quant-ph/9604034 DOI
- [13]
- N. J. Cerf and R. Cleve, “Information-theoretic interpretation of quantum error-correcting codes”, Physical Review A 56, 1721 (1997) arXiv:quant-ph/9702031 DOI
- [14]
- N. J. Cerf, “Entropic bounds on coding for noisy quantum channels”, Physical Review A 57, 3330 (1998) arXiv:quant-ph/9707023 DOI
- [15]
- P. Hayden, R. Jozsa, D. Petz, and A. Winter, “Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality”, Communications in Mathematical Physics 246, 359 (2004) arXiv:quant-ph/0304007 DOI
- [16]
- V. Korepin and J. Terilla, “Thermodynamic interpretation of quantum error correcting criterion”, (2004) arXiv:quant-ph/0202054
- [17]
- M.-D. Choi, D. W. Kribs, and K. Życzkowski, “Quantum error correcting codes from the compression formalism”, Reports on Mathematical Physics 58, 77 (2006) arXiv:quant-ph/0511101 DOI
- [18]
- M.-D. Choi, D. W. Kribs, and K. Zyczkowski, “Higher-Rank Numerical Ranges and Compression Problems”, (2006) arXiv:math/0511278
- [19]
- D. W. Kribs and S. Plosker, “Private quantum codes: introduction and connection with higher rank numerical ranges”, Linear and Multilinear Algebra 62, 639 (2013) arXiv:1407.1350 DOI
- [20]
- A. Tanggara, M. Gu, and K. Bharti, “Strategic Code: A Unified Spatio-Temporal Framework for Quantum Error-Correction”, (2024) arXiv:2405.17567
- [21]
- M. M. Wilde, Quantum Information Theory (Cambridge University Press, 2013) DOI
- [22]
- D. W. Kribs, A. Pasieka, and K. Zyczkowski, “Entropy of a quantum error correction code”, (2008) arXiv:0811.1621
- [23]
- C. Bény and O. Oreshkov, “General Conditions for Approximate Quantum Error Correction and Near-Optimal Recovery Channels”, Physical Review Letters 104, (2010) arXiv:0907.5391 DOI
- [24]
- G. Chiribella, M. Dall’Arno, G. M. D’Ariano, C. Macchiavello, and P. Perinotti, “Quantum error correction with degenerate codes for correlated noise”, Physical Review A 83, (2011) arXiv:1007.3655 DOI
- [25]
- S. Lloyd, “Capacity of the noisy quantum channel”, Physical Review A 55, 1613 (1997) arXiv:quant-ph/9604015 DOI
- [26]
- P. W. Shor, “The quantum channel capacity and coherent information”, 2002 (obtained from the MSRI Workshop on Quantum Computation website) URL
- [27]
- I. Devetak, “The private classical capacity and quantum capacity of a quantum channel”, (2004) arXiv:quant-ph/0304127
- [28]
- J. Watrous, The Theory of Quantum Information (Cambridge University Press, 2018) DOI
- [29]
- M. M. Wilde, Quantum Information Theory (Cambridge University Press, 2016) arXiv:1106.1445 DOI
- [30]
- M. Christandl and A. Müller-Hermes, “Fault-Tolerant Coding for Quantum Communication”, IEEE Transactions on Information Theory 70, 282 (2024) arXiv:2009.07161 DOI
- [31]
- W. Doeblin, “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 (1937)
- [32]
- A. Makur and J. Singh, “Doeblin Coefficients and Related Measures”, IEEE Transactions on Information Theory 70, 4667 (2024) arXiv:2309.08475 DOI
- [33]
- M. Yoshida, G. Kimura, T. Miyadera, H. Imai, and J. Cheng, “Solution to the mean king’s problem using quantum error-correcting codes”, Physical Review A 91, (2015) arXiv:1507.07072 DOI
- [34]
- L. Vaidman, Y. Aharonov, and D. Z. Albert, “How to ascertain the values ofsigmax,σy, andσzof a spin-1/2particle”, Physical Review Letters 58, 1385 (1987) DOI
- [35]
- J. P. Paz and W. H. Zurek, “Continuous error correction”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 355 (1998) DOI
- [36]
- J. P. Barnes and W. S. Warren, “Automatic Quantum Error Correction”, Physical Review Letters 85, 856 (2000) arXiv:quant-ph/9912104 DOI
- [37]
- C. Ahn, A. C. Doherty, and A. J. Landahl, “Continuous quantum error correction via quantum feedback control”, Physical Review A 65, (2002) arXiv:quant-ph/0110111 DOI
- [38]
- M. Sarovar and G. J. Milburn, “Continuous quantum error correction by cooling”, Physical Review A 72, (2005) arXiv:quant-ph/0501038 DOI
- [39]
- R. van Handel and H. Mabuchi, “Optimal error tracking via quantum coding and continuous syndrome measurement”, (2005) arXiv:quant-ph/0511221
- [40]
- J. F. Poyatos, J. I. Cirac, and P. Zoller, “Quantum Reservoir Engineering with Laser Cooled Trapped Ions”, Physical Review Letters 77, 4728 (1996) DOI
- [41]
- O. Oreshkov, “Continuous-time quantum error correction”, (2013) arXiv:1311.2485
- [42]
- J.-M. Lihm, K. Noh, and U. R. Fischer, “Implementation-independent sufficient condition of the Knill-Laflamme type for the autonomous protection of logical qudits by strong engineered dissipation”, Physical Review A 98, (2018) arXiv:1711.02999 DOI
- [43]
- J. Lebreuilly, K. Noh, C.-H. Wang, S. M. Girvin, and L. Jiang, “Autonomous quantum error correction and quantum computation”, (2021) arXiv:2103.05007
- [44]
- O. Oreshkov and T. A. Brun, “Continuous quantum error correction for non-Markovian decoherence”, Physical Review A 76, (2007) arXiv:0705.2342 DOI
- [45]
- S. Kawabata, “Information-theoretical approach to control of quantum-mechanical systems”, Physical Review A 68, (2003) arXiv:quant-ph/0409187 DOI
- [46]
- A. Lanka, S. Hegde, and T. A. Brun, “Optimizing continuous-time quantum error correction for arbitrary noise”, (2026) arXiv:2506.21707
- [47]
- L. Colmenarez, Z.-M. Huang, S. Diehl, and M. Müller, “Accurate optimal quantum error correction thresholds from coherent information”, Physical Review Research 6, (2024) arXiv:2312.06664 DOI
Page edit log
- Victor V. Albert (2023-07-24) — most recent
- Milan Tenn (2023-07-17)
- Milan Tenn (2023-07-07)
- Milan Tenn (2023-07-03)
- Milan Tenn (2023-06-28)
- Victor V. Albert (2022-07-15)
- Victor V. Albert (2022-03-18)
- Victor V. Albert (2021-12-09)
Cite as:
“Finite-dimensional quantum error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/qecc_finite
Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/qecc_finite.yml.