## 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.

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

Stabilizer codewords encoding a logical oscillator (i.e., CV quantum information) admit either a discrete or a continuous stabilizer group. The former, called oscillator-into-oscillator GKP codes, are obtained from multimode GKP codes by removing stabilizer generators for some of the modes. The latter encode information in hyperplanes and 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 nullifiers [3] or annihilators. An oscillator-into-oscillator stabilizer code encoding \(k\) logical modes into \(n\) physical modes is denoted as \([[n,k,d]]_{\mathbb{R}}\), where \(d\) is the code's distance.

## Protection

## Gates

## Parents

## Children

- Analog stabilizer code — Analog stabilizer codes are bosonic stabilizer codes with a continuous stabilizer group, corresponding to linear constraints on positions and momenta.
- Quantum lattice code — Quantum lattice codes are bosonic stabilizer codes with a countably infinite stabilizer group, corresponding to modular constraints on positions and momenta.

## Cousins

- Calderbank-Shor-Steane (CSS) stabilizer code — An oscillator stabilizer code admitting a set of generators such that each generator consists of either position or momentum operators is a CSS code.
- Group GKP code — The group-GKP construction encompasses all bosonic CSS codes. A single-mode qubit GKP code corresponds to the \(2\mathbb{Z}\subset\mathbb{Z}\subset\mathbb{R}\) group construction, and multimode GKP codes can be similarly described. An \([[n,k,d]]_{\mathbb{R}}\) analog stabilizer code corresponds to the \(\mathbb{R}^{ k_1} \subseteq \mathbb{R}^{ k_2} \subset \mathbb{R}^{n}\) group construction, where \(k=k_2/k_1\). GKP stabilizer codes for \(n\) modes correspond to subgroups \(\mathbb{Z}^m\) for \(m<n\).
- Number-phase code — Number-phase codewords span the joint right eigenspace of the \(N\)th power of the Susskind–Glogower phase operator and the bosonic rotation operator [5]. These operators no longer form a group since the phase operator is not unitary.
- Homological number-phase code — Homological number-phase codewords span the joint right eigenspace of powers of the non-unitary Susskind–Glogower phase operators and unitary bosonic rotation operators.

## References

- [1]
- R. L. Barnes, “Stabilizer Codes for Continuous-variable Quantum Error Correction”, (2004) arXiv:quant-ph/0405064
- [2]
- J. Bermejo-Vega and M. V. den Nest, “Classical simulations of Abelian-group normalizer circuits with intermediate measurements”, (2013) arXiv:1210.3637
- [3]
- M. Gu et al., “Quantum computing with continuous-variable clusters”, Physical Review A 79, (2009) arXiv:0903.3233 DOI
- [4]
- S. D. Bartlett and W. J. Munro, “Quantum Teleportation of Optical Quantum Gates”, Physical Review Letters 90, (2003) arXiv:quant-ph/0208022 DOI
- [5]
- A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020) arXiv:1901.08071 DOI

## Page edit log

- Victor V. Albert (2022-07-05) — most recent
- Victor V. Albert (2022-03-24)

## Cite as:

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