## Description

A code whose logical subspace is the joint eigenspace (usually with eigenvalue \(+1\)) of a set of commuting unitary 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. The stabilizer formalism is applicable to almost all quantum-code kingdoms; see list of stabilizer codes for a list of all stabilizer codes in the zoo.

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. Other qubit-based extensions, such as XS and XP stabilizer codes, relax the mutual commutation property. Still other extensions defined for Galois qudits include non-stabilizer codes.

## Parents

- Commuting-projector 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 commuting-projector Hamiltonian.
- Group-representation code — Stabilizer codes are group-representation codes since their projections are onto the trivial irrep of the stabilizer group [1].

## Children

## 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]
- A. Denys and A. Leverrier, “Multimode bosonic cat codes with an easily implementable universal gate set”, (2023) arXiv:2306.11621

## 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