Analog stabilizer code 


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


Protect against erasures of at most \(d-1\) modes, or arbitrarily large dispalcements on those modes. If an error operator does not commute with a nullifier, then that error is detectable. Protection of logical modes against small displacements cannot be done using only Gaussian resources [2,3] (see also [4,5]). There are no such restrictions for non-Gaussian noise [6].


Gaussian circuit applied to \(k\) modes storing logical information and \(n-k\) modes initialized in a position state.


Homodyne measurement of nullifiers yields real-valued syndromes, and recovery can be performed by displacements conditional on the syndromes.


One-sided device-independent QKD [7].




  • Oscillator-into-oscillator GKP code — Analog stabilizer codes protect logical modes against arbitrarily large displacements on a few modes, while oscillator-into-oscillator GKP codes protect a finite-dimensional logical space against sufficiently small displacements in any number of modes. Encoding in analog-stabilizer (oscillator-into-oscillator GKP) 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 [24], so oscillator-into-oscillator GKP codes can be thought of as analog stabilizer encodings utilizing non-Gaussian GKP resource states.
  • Modular-qudit stabilizer code — Prime-qudit stabilizer codes can be converted into analog stabilizer codes whose distance is at least as large as that of the original code [8].
  • 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.
  • Qudit cubic code — The qudit cubic code can be generalized to oscillators [9].


M. Gu et al., “Quantum computing with continuous-variable clusters”, Physical Review A 79, (2009) arXiv:0903.3233 DOI
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
C. Vuillot et al., “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019) arXiv:1810.00047 DOI
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
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
P. van Loock, “A note on quantum error correction with continuous variables”, (2008) arXiv:0811.3616
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
L. G. Gunderman, “Stabilizer Codes with Exotic Local-dimensions”, Quantum 8, 1249 (2024) arXiv:2303.17000 DOI
V. V. Albert, S. Pascazio, and M. H. Devoret, “General phase spaces: from discrete variables to rotor and continuum limits”, Journal of Physics A: Mathematical and Theoretical 50, 504002 (2017) arXiv:1709.04460 DOI
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Zoo Code ID: analog_stabilizer

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“Analog stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_analog_stabilizer, title={Analog stabilizer code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Analog stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.