Bosonic stabilizer code[1]


Also known as a continuous-variable (CV) stabilizer code. 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. 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 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 nullifiers [2] 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.


Protective properties can be delineated in terms of the nullifiers or displacements, and the most natural noise model for such codes is displacement noise. If an error operator does not commute with a stabilizer group element, then that error is detectable. Oscillator-into-oscillator stabilizer codes protect against erasures of a subset of modes, while GKP codes protect against sufficiently small displacements in any number of modes.




  • Group GKP code — The group-GKP construction encompasses all bosonic CSS codes. A singlemode 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 Pegg-Barnett phase operator and the bosonic rotation operator [3]. These operators no longer form a group since the phase operator is not unitary.


R. L. Barnes, “Stabilizer Codes for Continuous-variable Quantum Error Correction”, (2004) arXiv:quant-ph/0405064
M. Gu et al., “Quantum computing with continuous-variable clusters”, Physical Review A 79, (2009) arXiv:0903.3233 DOI
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
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Zoo Code ID: oscillator_stabilizer

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