## 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.

Since quantum information is encoded in quantum superpositions, an additional source of noise (not relevant to classical encodings) can affect the relative phase of such superpositions. Quantum error-correcting codes have to protect against such phase-flip noise while also protecting against conventional classical bit-flip noise. Codes are not expected to protect against both types of noise perfectly, and there is generally a tradeoff.

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| \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].

Pseudo-threshold (a.k.a. break-even point): The ultimate goal of error correction is to make sure that the logical error rate is greater than the underlying physical error rate. For a noise model parameterized by a single physical error rate \(p\), the pseudo-threshold or break-even point is the smallest \(p\) at which the logical error rate after error correction is equal to \(p\).

## Rate

## Decoding

## Notes

## Parents

- Approximate quantum error-correcting code (AQECC) — QAECCs correcting a noise channel exactly reduce to QECCs.
- Operator-algebra QECC (OAQECC) — An OAQECC which has no gauge structure (e.g., gauge qubits) and no block structure is a QECC.

## Children

- Group-based quantum code
- Asymmetric quantum code
- Block quantum code
- Holographic code
- Group-representation code — Group-representation code projections are onto an irrep of a subgroup of canonical or distinguished unitary operations on a Hilbert space. Removing the restriction to distinguished operations and allowing all operations, every code projection on an \(N\)-dim Hilbert space can be expressed as a projection onto the irrep formed by the code-preserving subgroup of \(U(N)\). The same idea holds when \(N\) is taken to infinity. In other words, while all codes are covariant w.r.t. some group, group-representation codes are covariant w.r.t. a canonical or distinguished subgroup.
- Hamiltonian-based code
- Finite-dimensional quantum error-correcting code
- Random quantum code
- Monolithic quantum code

## Cousins

- Error-correcting code (ECC) — Quantum information cannot be copied using a linear process [29], so one cannot send several copies of a quantum state through a channel as can be done for classical information. The Knill-Laflamme conditions can similarly be formulated for classical codes [30; Sec. 3], although they are not as widely as used as those for quantum codes.
- 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.
- 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.
- Hybrid QECC — A hybrid QECC storing no classical information reduces to a QECC. Conversely, any QECC can be converted into a hybrid QECC by using a portion of its logical subspace to store only classical information.
- Subsystem QECC — A subsystem QECC reduces to an ordinary (i.e., subspace) QECC when the gauge subsystem is trivial. 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.
- Quantum error-transmuting code (QETC) — QETCs are quantum codes which satisfy a generalization of the Knill-Laflamme conditions. QETCs for which the admissible logical error set consists solely of the identity are QECCs.
- \([[9,1,3]]\) Shor code — The Shor code is the first quantum error-correcting code.

## References

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## 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/edit/main/codes/quantum/properties/qecc.yml.