# Analog stabilizer code

## Description

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

Analog 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 analog stabilizer code admitting a set a set of nullifiers such that each nullifier consists of either position or momentum operators is called an analog CSS code.

## Protection

## Encoding

## Decoding

## Realizations

## Parents

## Children

- Analog surface code
- Braunstein five-mode code
- EA analog stabilizer code — EA analog stabilizer codes utilize additional ancillary modes in a pre-shared entangled state, but reduce to ordinary analog stabilizer codes when said modes are interpreted as noiseless physical modes.
- Homological bosonic code
- Lloyd-Slotine nine-mode code

## Cousin

- GKP-stabilizer code — Analog stabilizer codes protect logical modes against arbitrarily 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 analog-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 [3][4][2], so GKP-stabilizer codes can be thought of as analog 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) arXiv:0903.3233 DOI
- [2]
- J. Niset, J. Fiurášek, and N. J. Cerf, “No-Go Theorem for Gaussian Quantum Error Correction”, Physical Review Letters 102, (2009) arXiv:0811.3128 DOI
- [3]
- C. Vuillot et al., “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019) arXiv:1810.00047 DOI
- [4]
- J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian States with Gaussian Operations is Impossible”, Physical Review Letters 89, (2002) arXiv:quant-ph/0204052 DOI
- [5]
- G. Giedke and J. Ignacio Cirac, “Characterization of Gaussian operations and distillation of Gaussian states”, Physical Review A 66, (2002) arXiv:quant-ph/0204085 DOI
- [6]
- P. van Loock, “A note on quantum error correction with continuous variables”, (2008) arXiv:0811.3616
- [7]
- E. Culf, T. Vidick, and V. V. Albert, “Group coset monogamy games and an application to device-independent continuous-variable QKD”, (2022) arXiv:2212.03935

## Page edit log

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

## Cite as:

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