# Bosonic stabilizer code

## Description

Bosonic code whose codespace is defined as the common \(+1\) eigenspace of a group of mutually commuting displacement operators. Displacements form the stabilizers of the code, and have continuous eigenvalues, in contrast with the discrete set of eigenvalues of qubit stabilizers. As a result, exact codewords are non-normalizable, so approximate constructions have to be considered. A stabilizer code admitting a set a set of group generators such that each generator consists of either position or momentum operators is called a bosonic CSS code.

Stabilizer codewords encoding a finite-dimensional codespace admit a discrete infinite stabilizer group. Such qudit-into-oscillator stabilizer codes are GKP and multimode GKP codes.

Stabilizer codewords encoding a logical oscillator admit either a discrete or a continuous stabilizer group. The former, called GKP-stabilizer codes, are obtained from multimode GKP codes by removing stabilizer generators for some of the modes. The latter can be defined in terms of the continuous group's Lie algebra, i.e., as the common \(0\)-eigenvalue eigenspace of mutually commuting linear combinations of oscillator position and momentum operators called annihilators. An \(((n,k,d))_{\mathbb{R}}\) oscillator-into-oscillator stabilizer code is denoted as \([[n,k,d]]_{\mathbb{R}}\), where \(d\) is the code's distance.

## Protection

## Parents

## Children

- Homological bosonic code — Homological CV codes are bosonic CSS codes.
- Multi-mode GKP code — GKP codes defined on rectangular lattice are bosonic CSS codes, which more general lattices yield bosonic stabilizer codes.

## Cousin

- Group GKP code — The group-GKP construction encompasses all bosonic CSS codes. For example, a single-mode qubit GKP code corresponds to the \(2\mathbb{Z}\subset\mathbb{Z}\subset\mathbb{R}\) group construction. For another example, an \([[n,k,d]]_{\mathbb{R}}\) oscillator-into-oscillator stabilizer code corresponds to the \(\mathbb{R}^{\times k_1} \subseteq \mathbb{R}^{\times k_2} \subset \mathbb{R}^{\times n}\) group construction, where \(k=k_2/k_1\).

## Zoo code information

## Cite as:

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