Finite-dimensional quantum error-correcting code

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 \(U:\mathsf{H}_{K}\to\mathsf{H}_{N}\) 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 \(c\) [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 [2][3], 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: In a finite-dimensional Hilbert space, there are necessary and sufficient conditions for a code to successfully correct a set of errors. These are called the Knill-Laflamme conditions [2][4][5]. 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 \(i,j\),} \end{align} where \(c_{ij}\) can be arbitrary numbers.

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 \(\text{rank}(c_{ij}) < |\cal{E}|\).

Gates

Universal gate set cannot be transversal for any code that detects single-qubit errors due to Eastin-Knill theorem [6].

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 inverse of an encoder, which does not correct errors.Quantum machine-learning based decoders such as quantum convolutional neural networks [7] and quantum autoencoders [8].

Parent

Children

References

[1]
D. Gottesman. Surviving as a quantum computer in a classical world
[2]
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
[3]
C. H. Bennett et al., “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996). DOI; quant-ph/9604024
[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]
B. Eastin and E. Knill, “Restrictions on Transversal Encoded Quantum Gate Sets”, Physical Review Letters 102, (2009). DOI; 0811.4262
[7]
I. Cong, S. Choi, and M. D. Lukin, “Quantum convolutional neural networks”, Nature Physics 15, 1273 (2019). DOI; 1810.03787
[8]
David F. Locher, Lorenzo Cardarelli, and Markus Müller, “Quantum Error Correction with Quantum Autoencoders”. 2202.00555
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Zoo code information

Internal code ID: qecc_finite

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Zoo Code ID: qecc_finite

Cite as:
“Finite-dimensional quantum error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qecc_finite
BibTeX:
@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={https://errorcorrectionzoo.org/c/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. https://errorcorrectionzoo.org/c/qecc_finite

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/properties/qecc_finite.yml.