# Stabilizer code

## 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. Stabilizer codes have been defined for qubits, modular qudits, Galois qudits, and oscillators using their respective Pauli-type groups.

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 the qubit, modular-qudit, Galois-qudit, bosonic, and fermionic 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 immensely 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 qudits include non-stabilizer codes.

An important property of qubit and qudit stabilizer codes is the QLDPC property, which means (roughly) that working with them remains not too hard as number of qudits grows; these remain as the primary ingredients for a quantum memory.

## Parent

## Children

## Cousins

- Linear code — Linear (stabilizer) codes form a large and well-studied subset of all classical (quantum) codes because features such as decoding and level of protection are typically easier to determine than those of nonlinear (non-stabilizer) codes.
- 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.

## Page edit log

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

## Zoo code information

## Cite as:

“Stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/properties/stabilizer.yml.