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 \(U:\mathsf{H}_{K}\to\mathsf{H}_{N}\) with 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|\end{align} for some constant \(c\). A code is said to protect against or correct the errors \(\mathcal{E}\).

Equivalently, correction capability is determined by the quantum error-correction conditions [1][2]: for all \(|\psi\rangle,|\phi\rangle\in\mathsf{H}_{N}\) and all errors \(E_a,E_b\in{\mathcal{E}}\), \begin{align} \langle \psi | E^{\dagger}_a E_b |\phi \rangle = C_{ab} \langle \psi | \phi \rangle~, \end{align} where the coefficients \(C_{ab}\) do not depend on \(|\psi\rangle\) or \(|\phi\rangle\). A code that satisfies these conditions approximately, i.e., up to some small quantifiable error, is called an approximate error-correcting code.

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 \(\text{rank}(C_{ab}) < |\cal{E}|\).


Universal gate set cannot be transversal for any code that detects single-qubit errors due to Eastin-Knill theorem [3].


The operation \(\cal{D}\) in the definition of this code is called the decoder. However, the term decoder can sometimes be used for the inverse of an encoder, which does not correct errors.Quantum machine-learning based decoders such as quantum convolutional neural networks [4] and quantum autoencoders [5].



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Internal code ID: qecc_finite

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Cite as:
“Finite-dimensional quantum error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_qecc_finite, title={Finite-dimensional quantum error-correcting code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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E. Knill, R. Laflamme, and L. Viola, “Theory of Quantum Error Correction for General Noise”, Physical Review Letters 84, 2525 (2000). DOI; quant-ph/9604034
C. H. Bennett et al., “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996). DOI; quant-ph/9604024
B. Eastin and E. Knill, “Restrictions on Transversal Encoded Quantum Gate Sets”, Physical Review Letters 102, (2009). DOI; 0811.4262
I. Cong, S. Choi, and M. D. Lukin, “Quantum convolutional neural networks”, Nature Physics 15, 1273 (2019). DOI; 1810.03787
David F. Locher, Lorenzo Cardarelli, and Markus Müller, “Quantum Error Correction with Quantum Autoencoders”. 2202.00555

Cite as:

“Finite-dimensional quantum error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.