Finite-dimensional quantum error-correcting code 


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.


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 of the quantum error-correction conditions, which may admit infinite terms due to non-normalizability of ideal code states. 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 [24][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 Knill-Laflamme conditions can alternatively be expressed in terms of the complementary channel, or in an information-theoretic way via a data processing inequality [6][7; Eq. (29)]. They have been extended to sequences of multiple errors and rounds of correction [8].

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 encdoing 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) and then tracing out the system factor (instead of the second environmental factor) [9; 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 [10].


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*{(6)}\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 [11].


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 exacly correctable; see AQECC entry.

Code Capacity Threshold

Coherent information of the state under the action of a noise channel can be used to estimate the optimal threshold [12].





D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
E. Knill and R. Laflamme, “Theory of quantum error-correcting codes”, Physical Review A 55, 900 (1997) DOI
C. H. Bennett et al., “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996) arXiv:quant-ph/9604024 DOI
J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2012) DOI
B. Schumacher and M. A. Nielsen, “Quantum data processing and error correction”, Physical Review A 54, 2629 (1996) arXiv:quant-ph/9604022 DOI
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
A. Tanggara, M. Gu, and K. Bharti, “Strategic Code: A Unified Spatio-Temporal Framework for Quantum Error-Correction”, (2024) arXiv:2405.17567
M. M. Wilde, Quantum Information Theory (Cambridge University Press, 2013) DOI
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
J. Watrous, The Theory of Quantum Information (Cambridge University Press, 2018) DOI
L. Colmenarez et al., “Accurate optimal quantum error correction thresholds from coherent information”, (2024) arXiv:2312.06664
Page edit log

Your contribution is welcome!

on (edit & pull request)— see instructions

edit on this site

Zoo Code ID: qecc_finite

Cite as:
“Finite-dimensional quantum error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@incollection{eczoo_qecc_finite, title={Finite-dimensional quantum error-correcting code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:

Cite as:

“Finite-dimensional quantum error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.