Gottesman-Kitaev-Preskill (GKP) States

Updated 18 December 2025

GKP states are bosonic quantum codes that encode discrete logical information using lattice translations in continuous-variable phase space.
They are stabilized by displacement operators, enabling effective correction of small phase-space shifts to protect quantum data.
Experimental advances in circuit QED, optical systems, and neural network optimizations drive practical implementations of approximate GKP states.

The Gottesman-Kitaev-Preskill (GKP) states are bosonic quantum code states defined in continuous-variable (CV) Hilbert spaces, with the canonical purpose of encoding discrete-variable (qubit or qudit) logical information in harmonic oscillators. Originating from the seminal GKP code construction, these states feature a remarkable synergy between discrete stabilizer code concepts and CV quantum error correction: they are stabilized by displacement operators corresponding to lattice translations, permitting the correction of small phase-space shifts—precisely the dominant errors in bosonic channels. GKP states exist in both idealized forms (infinite squeezing, Dirac-comb structure) and physically realistic, finite-energy approximations. Their code properties, preparation complexity, stabilizer structure, error-correction performance, and experimental instantiations are central to modern efforts toward fault-tolerant quantum computing in both optical and circuit QED platforms, as well as in quantum communication.

1. Mathematical Structure and Stabilizer Formalism

GKP states utilize the algebra of the CV phase space, with canonical operators $q$ and $p$ satisfying $[q,p]=i$ (with $\hbar = 1$ throughout). The ideal square-lattice GKP code embeds a single logical qubit into a bosonic mode, leveraging the simultaneous +1 eigenspace of two commuting displacement (stabilizer) operators: $S_q = e^{-2i\sqrt{\pi}p}, \qquad S_p = e^{+i\sqrt{\pi}q}.$ The logical computational basis states are infinite Dirac combs in position: $\ket{0_L} \propto \sum_{n\in\mathbb{Z}} D_q(2n\sqrt{\pi})\ket{0}_q = \sum_{n\in\mathbb{Z}}\ket{x=2n\sqrt{\pi}}_q,$ with grid spacing $2\sqrt{\pi}$ , and logical $|1_L\rangle$ obtained by a half-grid shift $D_q(\sqrt{\pi})$ . The logical Pauli operators correspond to half-grid translations $X_L = e^{-i\sqrt{\pi}p}$ and $Z_L = e^{i\sqrt{\pi}q}$ , with Clifford group operations realized via Gaussian unitaries. These states are exactly periodic in both quadratures, providing the fundamental protection against small phase-space shift errors (Goldberg, 16 Dec 2025, Hosseinynejad et al., 5 Nov 2025, Dhara et al., 6 Jun 2024, Banic et al., 14 Apr 2025).

2. Approximate (Finite-Energy) GKP States and Physical Realizability

The ideal codewords are not normalizable, requiring infinite energy. Realistic GKP states are constructed by replacing each Dirac peak with a narrow Gaussian of width $\sigma$ and enveloping the infinite sum with a broad Gaussian of width $1/\kappa$ : $\psi^{\Delta,\kappa}_j(x) = N_j \sum_{s\in\mathbb{Z}} \exp\left(-\frac{(2s+j)^2\pi\kappa^2}{2}\right) \exp\left(-\frac{(x-(2s+j)\sqrt\pi)^2}{2\sigma^2}\right).$ One frequently sets $\kappa=\sigma$ and denotes the envelope squeezing by $s_\text{GKP} = -10\log_{10}\Delta^2$ , with threshold performance for error correction at $s_\text{GKP}\gtrsim9.75$ dB (Hosseinynejad et al., 5 Nov 2025, Erkilic et al., 2 Dec 2025). In phase space, these states manifest as two-dimensional combs modulated by Wigner functions that are sums of Gaussians (Banic et al., 14 Apr 2025). Alternative envelope models employ Fock damping or random-shift (Gaussian convolution) pictures, all capturing the essential physics of finite-energy GKP encodings (Hosseinynejad et al., 5 Nov 2025, Erkilic et al., 2 Dec 2025, Calcluth et al., 2022).

Generalizations such as the GGKP construction on the quantum torus replace the non-compact phase space $\mathbb{R}^2$ with a compactified, noncommutative torus $\mathbb{T}^2_{\theta_0}$ , leading to codewords expressed as Riemann-Theta functions that are exactly normalizable and orthogonal without infinite squeezing (Joseph et al., 20 Sep 2025).

3. Stabilizer Diagnostic Metrics and Fidelity Bounds

A customary method for certifying code quality is via stabilizer expectation values (SEVs): $s_q = |\langle S_q \rangle|, \quad s_p = |\langle S_p \rangle|,$ with $s_q = s_p = 1$ only in the +1 eigenspace (ideal codeword). However, these SEVs do not lower bound the actual fidelity $F(\rho)$ to the GKP code space. The relationship is strictly an upper bound: $F(\rho) \leq \frac{(s_q+1)(s_p+1)}{4},$ meaning large SEVs can only certify that $F$ is not larger than this value, but poor SEVs can rule out high-fidelity codewords (Goldberg, 16 Dec 2025). Counterexamples explicitly demonstrate that states with $s_q, s_p \approx 1$ can have arbitrarily small fidelity to any ideal GKP codeword. Thus, SEVs are insufficient for genuine code certification and must be complemented with direct logical tomography, Wigner-based analysis, or full state fidelity estimation.

4. Preparation Protocols: Circuit, Cavity QED, and Optical Platforms

Preparation of high-quality GKP states is central for practical deployment, with protocols spanning circuit QED (superconducting cavities), cavity-optomechanical, and all-optical implementations.

Circuit and Reservoir-Engineering: Periodic driving of SQUID-shunted oscillators can generate GKP Floquet states as ground or near-ground states of an effective time-dependent Hamiltonian $H_GKP = -\hbar J[\cos(2\sqrt{\pi}x)+\cos(2\sqrt{\pi}p)]$ , via adiabatic frequency ramps in realistic circuit QED hardware, achieving $>10$ dB squeezing rapidly and robustly (Kolesnikow et al., 2023). Reservoir-engineered autonomous stabilization protocols use sequential conditional displacements and unconditional ancilla resets to autonomously correct to the GKP code space, delivering experimental increases in logical lifetime due to error correction (Lachance-Quirion et al., 2023, Royer et al., 2020). Exact similarity-transformed stabilizer and dark-mode analyses enable nonperturbative error models and second-order Trotterized circuit designs for high performance under dissipation.
Heralded and Deterministic Cat-Breeding Optical Schemes: In all-optical architectures, GKP states may be bred by successively interfering squeezed-cat states on beamsplitters and conditioning on homodyne measurement outcomes. Iterating this process or using probabilistic normalization-free heralded post-selection achieves GKP states at and above the fault-tolerance threshold with only modest resource requirements (input squeezing $<3$ dB, single-photon detectors, and OPA gain $<0.7$ ). Both iterative and one-shot multi-photon heralding strategies are feasible, with practical success probabilities and high-fidelity ( $F>0.99$ ) output (Erkilic et al., 2 Dec 2025, Winnel et al., 2023).
Cavity-QED and Quantum Memory Interfaces: Cavity-mediated conditional displacement gates, leveraging the reflection of photons from a memory-coupled cavity, generate entanglement between GKP qubits and matter-based quantum memories. Performance is optimal at large cooperativity $C$ and high coupling efficiency $\zeta$ ; this enables high-fidelity bidirectional state transfer and cluster-state generation between photonic GKP and solid-state qubits (Dhara et al., 6 Jun 2024, Hastrup et al., 2021).
Random Walk and Modular Encodings: Schemes based on discretized random walks in phase space—implemented with simple bulk-optical components and polarization qubits—efficiently generate approximate GKP combs with controlled envelope width, offering a compact physical realization paradigm (Sakuldee, 13 Sep 2024). Time-frequency GKP encodings in correlated single-photon wavepackets have also been analyzed for robustness to both phase and photon loss errors (Descamps et al., 2023).
Neural Network and Optimized Code Designs: Recent methods employ neural networks to minimize the number of squeezed-coherent superpositions needed for robust GKP codes, outperforming Gaussian-envelope conventional codes in error-correction capability with only one-third the number of components (e.g., 7 terms vs. 21 for $9.55$ dB squeezing) (Zeng et al., 2 Nov 2024).

5. Complexity, Error Models, and Classical Simulability

Precise circuit-size bounds for preparing (approximate) GKP states have recently been established. Any protocol consuming the standard gate set (Gaussian, Clifford+T qubit, qubit-controlled displacements, homodyne, etc.) requires a number of operations at least $\Theta(\log[1/\kappa]+\log[1/\Delta])$ to achieve envelope width $\kappa$ and peak width $\Delta$ , with optimal heralded protocols achieving this scaling (Brenner et al., 25 Oct 2024). This quantifies the resource overhead for scalable CV architectures.

For error modeling, finite-energy GKP cluster and graph states can be described as coherent Gaussian ensembles over randomly displaced ideal codewords, with update rules (covariances and means) under error correction and measurement specified via classical symplectic transformations (Seshadreesan et al., 2021). The Gaussian random noise (GRN) model, where each codeword is drawn from a Gaussian-displaced ideal code, provides a simple analytic description, but with explicit deviations for conditional (homodyne-postselected) operations (Banic et al., 14 Apr 2025).

Classical simulatability is tractable for large classes of circuits involving GKP states under certain Gaussian operations and homodyne measurements, as the evolved measurement PDFs collapse to mixtures of Dirac-comb distributions, allowing efficient sampling. This extends efficient classical simulation beyond the Clifford-encoded GKP group (Calcluth et al., 2022).