# Quantum error-correcting code (QECC)

## Description

## 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| \tag*{(1)}\end{align} for some constant \(c\).

Equivalently, correction capability is determined by the Knill-Laflamme conditions, which may admit infinite terms due to non-normalizability of ideal code states in the case of codes with infinite-dimenional physical spaces. A code that satisfies these conditions approximately, i.e., up to some small quantifiable error, is called an approximate code. These conditions can also be formulated in terms of a dual Heisenberg picture, where correctability is checked for some algebra of observables [1].

## Rate

## Notes

## Parents

## Children

- Entanglement-assisted (EA) QECC — EA QECCs utilize additional ancillary subsystems in a pre-shared entangled state, but reduce to QECCs when said subsystems are interpreted as noiseless physical subsystems.
- Group-based quantum code
- Bosonic code
- Approximate quantum error-correcting code (AQECC)
- Block quantum code
- Dynamically-generated QECC
- Hamiltonian-based code
- Holographic code
- Finite-dimensional quantum error-correcting code
- Quantum Lego code
- Random quantum code
- Stabilizer code

## Cousins

- Error-correcting code (ECC)
- Metrological code — Metrological codes are logical-qubit codes that satisfy the Knill-Laflamme conditions conditions only partially, and codes that satisfy them fully are QECCs.
- 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.

## References

- [1]
- C. Bény, A. Kempf, and D. W. Kribs, “Quantum error correction on infinite-dimensional Hilbert spaces”, Journal of Mathematical Physics 50, 062108 (2009) arXiv:0811.0421 DOI
- [2]
- S. Lloyd, “Capacity of the noisy quantum channel”, Physical Review A 55, 1613 (1997) arXiv:quant-ph/9604015 DOI
- [3]
- Peter W. Shor, The quantum channel capacity and coherent information, 2002 (obtained from the MSRI Workshop on Quantum Computation website).
- [4]
- I. Devetak, “The private classical capacity and quantum capacity of a quantum channel”, (2004) arXiv:quant-ph/0304127
- [5]
- “Preface to the Second Edition”, Quantum Information Theory xi (2016) arXiv:1106.1445 DOI
- [6]
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2012) DOI
- [7]
- J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
- [8]
- D. Gottesman. Surviving as a quantum computer in a classical world
- [9]
- B. M. Terhal, “Quantum error correction for quantum memories”, Reviews of Modern Physics 87, 307 (2015) arXiv:1302.3428 DOI
- [10]
- J. Roffe, “Quantum error correction: an introductory guide”, Contemporary Physics 60, 226 (2019) arXiv:1907.11157 DOI
- [11]
- W. G. Unruh, “Maintaining coherence in quantum computers”, Physical Review A 51, 992 (1995) arXiv:hep-th/9406058 DOI
- [12]
- “Is quantum mechanics useful?”, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences 353, 367 (1995) DOI

## Page edit log

- Victor V. Albert (2022-07-30) — most recent
- Philippe Faist (2022-07-15)
- Victor V. Albert (2022-01-03)

## 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/properties/qecc.yml.