## Description

A code whose logical subspace is the joint eigenspace (usually with eigenvalue \(+1\)) of a set of commuting unitary Pauli-type operators forming the code's stabilizer group. They can be block codes defined of tensor-product spaces of qubits or qudits, or non-block codes defined on single sufficiently large Hilbert spaces such as bosonic modes or group spaces.

The coding theory motivation for stabilizer codes came from linear binary codes, whose codewords form a closed subspace in the space of binary strings. Stabilizer codes extend this property, in various ways, to quantum error correction. Stabilizer codes can be defined succinctly using the stabilizer group generators and without explicitly writing out a basis of codewords. The stabilizer formalism is applicable to almost all quantum-code kingdoms; see list of stabilizer codes for a list of all stabilizer codes.

Stabilizer codes were originally defined for qubits, where the relevant commuting operators are tensor products of Pauli matrices. The Pauli stabilizer structure is useful in providing standardized encoding, gates, decoding, and performance bounds. Elements of this structure remain in qudit extensions, in particular for prime-dimensional modular qudits and Galois qudits. Infinite-dimensional Pauli-type bases yield the bosonic stabilizer and rotor stabilizer codes.

Extensions of the stabilizer formalism, such as XS and XP stabilizer codes, relax the mutual commutation property. Other extensions, such as CWS and union stabilizer codes, enlarge the codespace by re-assigning error words as codewords.

## Protection

## Decoding

## Parents

- Commuting-projector Hamiltonian code — Codespace is the ground-state space of the code Hamiltonian, which consists of an equal linear combination of stabilizer generators and which can be made into a frustration-free commuting-projector Hamiltonian.
- Frustration-free Hamiltonian code — Codespace is the ground-state space of the code Hamiltonian, which consists of an equal linear combination of stabilizer generators and which can be made into a frustration-free commuting-projector Hamiltonian.
- Knill code — Stabilizer codes are Knill codes whose nice error basis is either the Pauli strings, modular-qudit Pauli strings, Galois-qudit Pauli strings, oscillator displacement operators, or rotor generalized Pauli strings.

## Children

- Graph quantum code — Graph quantum codes are a subset of stabilizer codes over \(G\)-valued qudits for Abelian \(G\) [1]. Any stabilizer code over Abelian \(G\) is locally equivalent to a graph quantum code [1] (see also [2,3]).
- Rotor stabilizer code
- Bosonic stabilizer code
- Calderbank-Shor-Steane (CSS) stabilizer code
- Quantum low-weight check (QLWC) code
- Random stabilizer code
- Modular-qudit stabilizer code
- Galois-qudit stabilizer code

## Cousin

- Majorana stabilizer code — Majorana stabilizer codes are useful for Majorana-based architectures, where the degrees of freedom are electrons, and the notion of locality is different than all other code kingdoms.

## References

- [1]
- D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
- [2]
- M. Van den Nest, J. Dehaene, and B. De Moor, “Graphical description of the action of local Clifford transformations on graph states”, Physical Review A 69, (2004) arXiv:quant-ph/0308151 DOI
- [3]
- M. Grassl, A. Klappenecker, and M. Rotteler, “Graphs, quadratic forms, and quantum codes”, Proceedings IEEE International Symposium on Information Theory, arXiv:quant-ph/0703112 DOI

## Page edit log

- Victor V. Albert (2022-04-19) — most recent

## Cite as:

“Stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stabilizer