Quantum error-correcting code (QECC)

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 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.

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| \end{align} for some constant \(c\).

Equivalently, correction capability is determined by of the quantum error-correction conditions [1][2], 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.

Parent

Children

Cousins

  • Entanglement-assisted (EA) QECC — Entanglement-assisted QECCs are QECCs utilizing pre-shared entanglement.
  • Subsystem quantum error-correcting code — A subsystem code reduces to an ordinary error-correcting code when the gauge subsystem is trivial, \(\mathsf{B} = \mathbb{C}\). 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.

Zoo code information

Internal code ID: qecc

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: qecc

Cite as:
“Quantum error-correcting code (QECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qecc
BibTeX:
@incollection{eczoo_qecc, title={Quantum error-correcting code (QECC)}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qecc} }
Permanent link:
https://errorcorrectionzoo.org/c/qecc

References

[1]
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
[2]
C. H. Bennett et al., “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996). DOI; quant-ph/9604024

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/tree/main/codes/quantum/qecc.yml.