Continuous Variable Stabilizer States

• Continuous variable stabilizer states are infinite-dimensional quantum states defined by commuting displacement operators, establishing a robust framework for error-correcting codes.
• Gottesman–Kitaev–Preskill (GKP) codes encode logical qubits into bosonic modes via grid superpositions, using finitely squeezed states to exponentially suppress logical errors.
• Hybrid and topological CV stabilizer codes leverage oscillator–qudit entanglement and lattice structures, enabling scalable measurement-based quantum computation and fault-tolerant error correction.

Continuous variable (CV) stabilizer states are quantum states defined within infinite-dimensional Hilbert spaces—typically those of bosonic modes such as electromagnetic field modes—that generalize the stabilizer formalism of discrete-variable (qubit or qudit) systems. The CV stabilizer approach exploits the group structure of continuous displacement (Weyl) operators to define code spaces, logical operators, and error-correcting codes, enabling robust quantum information storage and processing. Prominent examples include Gottesman-Kitaev-Preskill (GKP) codes, hybrid oscillator-qudit codes, and CV topological codes, each leveraging different mathematical and physical constructs to realize fault-tolerant quantum protocols across both physical and hybrid (oscillator-discrete) architectures (Hosseinynejad et al., 5 Nov 2025, Chakraborty et al., 6 Aug 2025, Fuente et al., 7 Nov 2024).

1. Algebraic Structure of Continuous Variable Stabilizer States

CV stabilizer states arise from the action of Abelian subgroups of the displacement operator group on bosonic Hilbert spaces. For a single mode, the displacement ("Pauli") operators are $X(s) = e^{-is \hat{p}}$, $Z(t) = e^{it \hat{x}}$ with canonical commutation $[\hat{x}, \hat{p}] = i$ (setting $\hbar = 1$). These obey the Weyl commutation relation $X(s)Z(t) = e^{-ist}Z(t)X(s)$ (Fuente et al., 7 Nov 2024).

A CV stabilizer group is generated by a set of mutually commuting displacement operators $g_i = X(s_i)Z(t_i)$, typically forming a lattice in phase space. The stabilizer code space is defined as the simultaneous $+1$ eigenspace of all $g_i$, generalizing the discrete Pauli stabilizer code structure to the continuum. The set of displacement operators commuting with the stabilizer group forms the logical operator algebra. The quotient of the commutant by the stabilizer group, $C(S)/S$, characterizes the degrees of freedom encoded—these can be continuous (rotors) or discrete (qudits) depending on the stabilizer lattice geometry (Fuente et al., 7 Nov 2024, Chakraborty et al., 6 Aug 2025).

2. Gottesman-Kitaev-Preskill (GKP) Codes and Normalizable GKP States

GKP codes encode logical qubits into the infinite-dimensional Hilbert space of a bosonic mode using grid superpositions of position eigenstates. In the ideal square-lattice encoding,

$$|0_L\rangle = \sum_{s\in\mathbb{Z}} |q=2s\sqrt{\pi}\rangle, \quad |1_L\rangle = \sum_{s\in\mathbb{Z}} |q=(2s+1)\sqrt{\pi}\rangle$$

These states are stabilized by commuting displacement operators $S_q = e^{2i\sqrt{\pi}\hat{p}}$, $S_p = e^{-2i\sqrt{\pi}\hat{q}}$. Logical Pauli operators are implemented by half-displacements: $X_L = e^{-i\sqrt{\pi}\hat{p}}$, $Z_L = e^{i\sqrt{\pi}\hat{q}}$ (Hosseinynejad et al., 5 Nov 2025).

Physical GKP states must be normalizable—ideal states require infinite energy—so a Gaussian envelope of width $\Delta$ is applied to the comb peaks; in the Fock basis, this corresponds to $$|\widetilde{s_L}\rangle = N_\beta e^{-\beta \hat{n}}|s_L\rangle$$ with $\Delta^2 \simeq \frac{1}{2}e^{2\beta} - 1$. Finitely squeezed GKP states ($\Delta \ll 1$) maintain sharply separated, narrow peaks, and the envelope entropy introduces exponentially suppressed logical errors scaling as $e^{-\pi/(2\Delta^2)}$ (Hosseinynejad et al., 5 Nov 2025).

3. Measurement-Based Quantum Computation with CV Stabilizer States

CV stabilizer states, especially finitely-squeezed GKP states, provide a framework for measurement-based quantum computation using only Gaussian operations (linear optics, phase shifts, squeezing) and homodyne detection. Clifford operations (Hadamard, phase, Pauli) are implemented by teleporting logical states through ancilla GKP wires via Gaussian circuits; e.g., entangling two modes with a continuous-variable controlled-phase (CV-CZ) gate and q-homodyne measurement enables Pauli-axis projections conditioned on measurement outcomes (Hosseinynejad et al., 5 Nov 2025).

For universal quantum computation, injection of non-Clifford (“magic”) states is necessary. Finite squeezing allows for probabilistic, heralded projection onto non-Pauli eigenstates—an outcome highly sensitive to the rotation angle in the teleportation circuit. For experimentally relevant squeezing (e.g., $\Delta\approx0.04$, ~15 dB), the probability of high-fidelity magic state generation (fidelity $F\geq0.96$) exceeds $40\%$, sufficient for magic-state distillation. All gate errors decay exponentially with inverse squeezing, maintaining Clifford fidelity $F_{cl}>0.999$ for $\Delta\lesssim0.2$ (Hosseinynejad et al., 5 Nov 2025). Thus, realistic GKP stabilizer states, via Gaussian protocols and homodyne readout, act as a resource for fault-tolerant universal computation.

4. Hybrid Oscillator-Qudit Stabilizer States

The CV stabilizer formalism is extended to hybrid quantum lattices comprising both continuous (oscillator) and discrete (qudit) degrees of freedom. The configuration space is $$M = \mathbb{R}^p \times \mathbb{Z}_{c_1}\times\cdots\times\mathbb{Z}_{c_k}$$, and the phase space $G = M \times \widehat{M}$ supports displacement operators and associated symplectic structure $J = J_{\mathrm{cv}}\oplus J_{\mathrm{dv}}$, with stabilizers embedded as integer lattices $T:\mathbb{Z}^{2n}\rightarrow G$ satisfying $T^TJ T =\Theta$ (Gram matrix) (Chakraborty et al., 6 Aug 2025).

Simple hybrid stabilizer (“LCA”) states are constructed by conditional displacements between a GKP state and a qudit, generating entangled superpositions indexed by both oscillator and discrete variables: $$|\mathrm{LCA}\rangle = \sum_{\ell\in\mathbb{Z}} |\ell\sqrt{2\pi/c}\rangle_x\otimes|-\ell\rangle_Z$$ Symplectic (Gaussian-Clifford) operations on such hybrid systems factorize, acting as tensor products on oscillator and qudit spaces, and thus cannot generate oscillator–qudit entanglement. Resourcefulness arises from non-symplectic gates, such as controlled displacements (Chakraborty et al., 6 Aug 2025).

Logical operator and code construction uses the theory of non-commutative tori and Morita equivalence. Starting from the stabilizer Gram matrix $\Theta$, one derives the logical algebra from its Morita dual $\Theta^\perp$ via an $\mathrm{SO}(2n,2n|\mathbb{Z})$ action, allowing systematic generation of hybrid error-correcting codes.

5. Topological and Intrinsic CV Stabilizer Codes

Continuous-variable stabilizer codes can be constructed in topologically nontrivial settings, generalizing both qudit surface codes and homological rotor codes. For example, the toric–GKP code layers single-mode GKP stabilizers on each edge of a lattice, then imposes toric-code–type stabilizers (vertex $A_v$ and plaquette $B_p$ operators) on the grid: $$A_v = \prod_{e\supset v} X_e(\sqrt{\pi}\,\sigma_{v,e}), \quad B_p = \prod_{e\in \partial p} Z_e(\sqrt{\pi}\,\sigma_{p,e})$$ The resulting stabilizer code space encodes, e.g., two logical CVs or qubits on a torus, with logical operators formed by loop products of $X(s)$ and $Z(t)$ along noncontractible cycles (Fuente et al., 7 Nov 2024).

Further, codes such as those built by boson condensation from a parent $\mathbb{R}$ gauge theory realize anyon theories (e.g., $U(1)_{2n}\times U(1)_{-2m}$ Chern-Simons) unattainable by qudit stabilizer models. Condensation is implemented via hopping (e.g., $C_e(2\pi,n)$) and projecting to subspaces invariant under bosonic subgroups. Hamiltonians for such codes can be made gapped with appropriate quadratic perturbations, while maintaining a topologically protected code space.

A key conjecture holds that these boson-condensation–engineered CV codes are "intrinsic" to the oscillator Hilbert space platform, in the sense that their topological orders—characterized by chiral central charges and Witt classes—cannot be obtained by concatenating qudit codes with GKP encodings. This suggests new fault-tolerant architectures exploiting infinite-dimensional collective properties (Fuente et al., 7 Nov 2024).

6. Error-Correcting Properties and Scalability

Error-correcting capabilities in CV stabilizer codes are characterized by code distances set by the phase-space lattice cell size. For instance, in hybrid LCA codes the minimum correctable displacement in either $\hat{x}$ or $\hat{p}$ is determined by the combined cell geometry; e.g., for a $(c,d)$ LCA code with $c=3$, $d=2$, the minimal distance is $d_{\mathrm{LCA}} = \sqrt{6\pi}$, protecting against position/momentum shifts and arbitrary single-qudit Pauli errors (Chakraborty et al., 6 Aug 2025).

Topological CV codes exploit global constraints of the phase-space stabilizer group, enabling scalable protection via delocalized encoded logicals. Unlike concatenated models (GKP to qudit code), intrinsic CV codes collectively utilize the full oscillator manifold, offering routes to lower overhead and fault-tolerant logicals not limited by finite-dimensional intermediate encodings (Fuente et al., 7 Nov 2024).

7. Connections and Outlook

Continuous-variable stabilizer states unify several advanced architectures for quantum computation, quantum memories, and quantum error correction. Their mathematical formalism draws on group theory, symplectic geometry, topological order, and non-commutative geometry. Experimentally relevant approximations—such as finitely squeezed GKP states—are not deficiencies but essential resources providing high-fidelity Clifford operations and heralded non-Clifford gates within measurement-based models (Hosseinynejad et al., 5 Nov 2025).

The interplay between intrinsic topological codes, hybrid oscillator-qudit constructions, and universal measurement-based computation points to a landscape of scalable quantum protocols exploiting both the infinite-dimensional nature of bosonic systems and discrete variable robustness. A plausible implication is that these approaches obviate the need for intermediate finite-dimensional encodings in future fault-tolerant architectures.