Here is a list of bosonic stabilizer codes.
Code Description
Analog stabilizer code An 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 surface code An analog CSS version of the Kitaev surface code realizing a phase of 2D \(\mathbb{R}\) gauge theory.
Analog-cluster-state code A code based on a continuous-variable (CV), or analog, cluster state. Such a state can be used to perform MBQC of logical modes, which substitutes the temporal dimension necessary for decoding a conventional code with a spatial dimension. The exact analog cluster state is non-normalizable, so approximate constructs have to be considered.
Bosonic 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 groups can contain discrete or continuous subgroups and can admit logical qudit and/or oscillator logical subspaces.
Compactified \(\mathbb{R}\) gauge theory code An integer-homology bosonic CSS code realizing 2D \(U(1)\) gauge theory on bosonic modes. The code can be obtained from the analog surface code by condensing certain anyons [1]. This results in a pinning of each mode to the space of periodic functions, which make up a physical rotor, and can be thought of as compactification of the 2D \(\mathbb{R}\) gauge theory phase realized by the analog surface code.
Concatenated GKP code A concatenated code whose outer code is a GKP code. In other words, a bosonic code that can be thought of as a concatenation of an arbitrary inner code and another bosonic outer code. Most examples encode physical qubits of an inner stabilizer code into the square-lattice GKP code.
GKP CV-cluster-state code Cluster-state code can consists of a generalized analog cluster state that is initialized in GKP (resource) states for some of its physical modes. Alternatively, it can be thought of as an oscillator-into-oscillator GKP code whose encoding consists of initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i V_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(V_{jk}\). Provides a way to perform fault-tolerant MBQC, with the required number \(n-k\) of GKP-encoded physical modes determined by the particular protocol [2–5].
GKP-surface code A concatenated code whose outer code is a GKP code and whose inner code is a toric surface code [6], rotated surface code [4,7–10], or XZZX surface code [11].
Gottesman-Kitaev-Preskill (GKP) code Quantum lattice code for a non-degenerate lattice, thereby admitting a finite-dimensional logical subspace. Codes on \(n\) modes can be constructed from lattices with \(2n\)-dimensional full-rank Gram matrices \(A\).
Hayden-Nezami-Salton-Sanders bosonic code An \([[n,1]]_{\mathbb{R}}\) analog CSS code defined using homological structres associated with an \(n-1\) simplex. Relevant to the study of spacetime replication of quantum information [12].
Hexagonal GKP code Single-mode GKP qudit-into-oscillator code based on the hexagonal lattice. Offers the best error correction against displacement noise in a single mode due to the optimal packing of the underlying lattice.
Integer-homology bosonic CSS code An oscillator stabilizer code whose physical modes have been restricted, via a single GKP stabilizer, from the space of functions on the real line to the space of periodic functions. This restriction effectively realizes a rotor on each physical mode, allowing one to construct homological rotor codes out of displacement stabilizer groups. The stabilizer group is continuous, but contains discrete components in the form of the single-mode GKP stabilizers. The homology group of the logical operators has a torsion component because the chain complexes are defined over the ring of integers, which yields codes with finite logical dimension.
NTRU-GKP code Multi-mode GKP code whose underlying lattice is utilized in variations of the NTRU cryptosystem [13]. Randomized constructions yield constant-rate GKP code families whose largest decodable displacement length scales as \(O(\sqrt{n})\) with high probability.
Oscillator-into-oscillator GKP code Multimode GKP code with an infinite-dimensional logical space. Can be obtained by considering an \(n\)-mode GKP code with a finite-dimensional logical space, removing stabilizers such that the logical space becomes infinite dimensional, and applying a Gaussian circuit.
Quantum lattice code Bosonic stabilizer code on \(n\) bosonic modes whose stabilizer group is an infinite countable group of oscillator displacement operators which implement lattice translations in phase space.
Square-lattice GKP code Single-mode GKP qudit-into-oscillator code based on the rectangular lattice. Its stabilizer generators are oscillator displacement operators \(\hat{S}_q(2\alpha)=e^{-2i\alpha \hat{p}}\) and \(\hat{S}_p(2\beta)=e^{2i\beta \hat{x}}\). To ensure \(\hat{S}_q(2\alpha)\) and \(\hat{S}_p(2\beta)\) generate a stabilizer group that is Abelian, there is a constraint that \(\alpha\beta=2q\pi\) where \(q\) is an integer denoting the logical dimension.
\(D_4\) hyper-diamond GKP code Two-mode GKP qudit-into-oscillator code based on the \(D_4\) hyper-diamond lattice.
\(U(1)_{2n} \times U(1)_{-2m}\) Chern-Simons GKP code A non-CSS multimode GKP code defined on a 2D mode lattice that encodes a qudit logical space and whose excitations are characterized by the \(U(1)_{2n} \times U(1)_{-2m}\) Chern-Simons theory. The code can be obtained from the analog surface code by condensing certain anyons [1].
\([[5,1,3]]_{\mathbb{R}}\) Braunstein five-mode code An analog stabilizer version of the five-qubit perfect code, encoding one mode into five and correcting arbitrary errors on any one mode.
\([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code An analog stabilizer version of Shor's nine-qubit code, encoding one mode into nine and correcting arbitrary errors on any one mode.

References

[1]
J. C. M. de la Fuente, T. D. Ellison, M. Cheng, and D. J. Williamson, “Topological stabilizer models on continuous variables”, (2024) arXiv:2411.04993
[2]
N. C. Menicucci, “Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States”, Physical Review Letters 112, (2014) arXiv:1310.7596 DOI
[3]
J. E. Bourassa et al., “Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer”, Quantum 5, 392 (2021) arXiv:2010.02905 DOI
[4]
K. Fukui, A. Tomita, A. Okamoto, and K. Fujii, “High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction”, Physical Review X 8, (2018) arXiv:1712.00294 DOI
[5]
I. Tzitrin, T. Matsuura, R. N. Alexander, G. Dauphinais, J. E. Bourassa, K. K. Sabapathy, N. C. Menicucci, and I. Dhand, “Fault-Tolerant Quantum Computation with Static Linear Optics”, PRX Quantum 2, (2021) arXiv:2104.03241 DOI
[6]
C. Vuillot, H. Asasi, Y. Wang, L. P. Pryadko, and B. M. Terhal, “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019) arXiv:1810.00047 DOI
[7]
K. Noh and C. Chamberland, “Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code”, Physical Review A 101, (2020) arXiv:1908.03579 DOI
[8]
M. V. Larsen, C. Chamberland, K. Noh, J. S. Neergaard-Nielsen, and U. L. Andersen, “Fault-Tolerant Continuous-Variable Measurement-based Quantum Computation Architecture”, PRX Quantum 2, (2021) arXiv:2101.03014 DOI
[9]
K. Noh, C. Chamberland, and F. G. S. L. Brandão, “Low-Overhead Fault-Tolerant Quantum Error Correction with the Surface-GKP Code”, PRX Quantum 3, (2022) arXiv:2103.06994 DOI
[10]
M. Lin, C. Chamberland, and K. Noh, “Closest Lattice Point Decoding for Multimode Gottesman-Kitaev-Preskill Codes”, PRX Quantum 4, (2023) arXiv:2303.04702 DOI
[11]
J. Zhang, Y.-C. Wu, and G.-P. Guo, “Concatenation of the Gottesman-Kitaev-Preskill code with the XZZX surface code”, Physical Review A 107, (2023) arXiv:2207.04383 DOI
[12]
P. Hayden and A. May, “Summoning information in spacetime, or where and when can a qubit be?”, Journal of Physics A: Mathematical and Theoretical 49, 175304 (2016) arXiv:1210.0913 DOI
[13]
J. Hoffstein, J. Pipher, and J. H. Silverman, “NTRU: A ring-based public key cryptosystem”, Lecture Notes in Computer Science 267 (1998) DOI