# Gaussian stabilizer code

## Description

Also known as an analog stabilizer code. Oscillator-into-oscillator stabilizer code encoding \(k\) logical modes into \(n\) physical modes. An \(((n,k,d))_{\mathbb{R}}\) Gaussian stabilizer code is denoted as \([[n,k,d]]_{\mathbb{R}}\), where \(d\) is the code's distance.

Gaussian stabilizer codes admit continuous stabilizer group of displacements. This group can equivalently be defined in terms of its Lie algebra. The codespace is equivalently the common \(0\)-eigenvalue eigenspace of the Lie algebra generators, which are mutually commuting linear combinations of oscillator position and momentum operators called nullifiers [1] or annihilators. A Gaussian stabilizer code admitting a set a set of nullifiers such that each nullifier consists of either position or momentum operators is called a Gaussian CSS code.

## Protection

## Encoding

## Decoding

## Parents

## Children

## Cousin

- GKP-stabilizer code — Gaussian stabilizer codes protect logical modes against artbirarily large displacements on a few modes, while GKP-stabilizer codes protect a finite-dimensional logical space against sufficiently small displacements in any number of modes. Encoding in Gaussian-stabilizer (GKP-stabilizer) codes can be done by a Gaussian operation acting on a tensor product of an arbitrary state in the first mode and position states (GKP states) on the remaining modes. Protection of logical modes against small displacements cannot be done using only Gaussian resources [2][3][4], so GKP-stabilizer codes can be thought of as Gaussian stabilizer encodings utilizing non-Gaussian GKP resource states.

## References

- [1]
- M. Gu et al., “Quantum computing with continuous-variable clusters”, Physical Review A 79, (2009). DOI; 0903.3233
- [2]
- C. Vuillot et al., “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019). DOI; 1810.00047
- [3]
- J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian States with Gaussian Operations is Impossible”, Physical Review Letters 89, (2002). DOI; quant-ph/0204052
- [4]
- J. Niset, J. Fiurášek, and N. J. Cerf, “No-Go Theorem for Gaussian Quantum Error Correction”, Physical Review Letters 102, (2009). DOI; 0811.3128

## Zoo code information

## Cite as:

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