GKP Qubit (Gottesman-Kitaev-Preskill Code)

Figure

Description

The GKP (Gottesman-Kitaev-Preskill) code, proposed in 2001, encodes a qubit in the position/momentum space of a harmonic oscillator using grid states — superpositions of periodically spaced position (or momentum) eigenstates. The ideal logical states are:

$|0_L⟩\propto {\sum }_{s=-\infty }^{\infty }|q=2s\sqrt{\pi }⟩,|1_L⟩\propto {\sum }_{s=-\infty }^{\infty }|q=(2s+1)\sqrt{\pi }⟩$

These states have a comb-like structure in position space with spacing $2\sqrt{\pi }$ and are displaced by $\sqrt{\pi }$ between $|0_L⟩$ and $|1_L⟩$. The key property is that small displacement errors $e^{i\xi \hat{p}}$ or $e^{-i\eta \hat{q}}$ (with $|\xi |,|\eta |<\sqrt{\pi }/2$) can be detected and corrected by measuring the stabilizers $e^{2i\sqrt{\pi }\hat{p}}$ and $e^{-2i\sqrt{\pi }\hat{q}}$.

In practice, ideal GKP states have infinite energy. Physical GKP states replace the delta-function comb with finite-width peaks under a finite envelope. A parameter $\Delta$ is often used for the peak width, but its precise definition varies across papers. With the symmetric convention used below, 10 dB of peak squeezing corresponds to $\Delta \approx 0.224$, not $\Delta \approx 0.5$; fault-tolerance thresholds depend on the surrounding architecture and noise model rather than on one universal value of $\Delta$.

GKP states have been realized in trapped-ion motional modes, superconducting microwave cavities, and integrated photonics. Superconducting-cavity experiments have demonstrated beyond-break-even GKP memories, while trapped-ion experiments have demonstrated a universal logical gate set including a two-GKP-qubit entangling gate.

Hamiltonian

GKP states are eigenstates of the stabilizer operators:

$S_q=e^{2i\sqrt{\pi }\hat{p}},S_p=e^{-2i\sqrt{\pi }\hat{q}}$

with eigenvalue +1. The logical Pauli operators are:

$\bar{X}=e^{i\sqrt{\pi }\hat{p}},\bar{Z}=e^{-i\sqrt{\pi }\hat{q}}$

For a common symmetric finite-energy convention with peak variance ${\Delta }^2$, the logical-zero position wavefunction is

${\psi }_{0,\Delta }(q)\propto {\sum }_{s\in \mathbb{Z}}{e}^{-2\pi {\Delta }^2{s}^2}\exp \left[-\frac{(q-2s\sqrt{\pi }{)}^2}{2{\Delta }^2}\right],$

with ${\psi }_{1,\Delta }(q)={\psi }_{0,\Delta }(q-\sqrt{\pi })$ up to the corresponding shifted envelope. This keeps the $2\sqrt{\pi }$ logical-comb spacing explicit and avoids mixing the $|0_L⟩$ and $|1_L⟩$ peak sets.

Relative to vacuum quadrature variance $1/2$, this convention gives ${\text{sq}}_{\text{dB}}=-10{\log }_{10}(2{\Delta }^2)$.

Motivation

GKP codes convert sufficiently small shifts in either phase-space quadrature into discrete syndrome information. Photon loss, thermal noise, and dephasing do not become literal bounded shifts in every regime, but GKP recovery can digitize their leading effects into correctable displacement and Pauli errors. The code is especially attractive because Gaussian operations implement its Clifford gates and because beyond-break-even GKP memories have been demonstrated in superconducting circuits.

Experimental Status

First realization in trapped-ion motional mode — Flühmann et al. (2019):

• Prepared approximate GKP states in the motional mode of a trapped ${}^{40}{\text{Ca}}^{+}$ ion
• Demonstrated stabilizer measurements and small displacement error correction
• Verified grid structure via Wigner function tomography

Superconducting cavity realization — Campagne-Ibarcq et al. (2020):

• GKP states prepared in a 3D superconducting microwave cavity using a transmon ancilla
• Demonstrated SBS (small-Big-small) displacement gates with 99.7% single-qubit gate fidelity
• First real-time GKP error correction in a circuit QED platform

Beyond-break-even QEC — Sivak et al. (2023):

• Pauli-averaged logical coherence gain $G=2.27\pm 0.07$ over the best physical component
• First demonstration of beyond-break-even quantum error correction
• Used real-time feedback with optimized control via reinforcement learning

Universal trapped-ion logical gates — Matsos et al. (2025):

• Deterministic finite-energy GKP control in two motional modes of one trapped ion
• Logical single-qubit process fidelity up to 0.960
• First two-GKP-qubit entangling gate, with CZ process fidelity 0.680, and direct Bell-state fidelity 0.842

Higher-dimensional and photonic GKP states — Brock et al. and Larsen et al. (2025):

• Beyond-break-even GKP qutrit and ququart memories with gains $1.82\pm 0.03$ and $1.87\pm 0.03$
• Integrated 1550-nm photonic GKP source with at least four resolved peaks in each quadrature and a $3\times 3$ lattice of negative Wigner regions; the reported symmetric effective squeezing was only $0.62\pm 0.02$ dB, so the state was not yet fault tolerant

Planar superconducting GKP QEC — Jha et al. (2026, preprint):

• Replaced the centimeter-scale 3D cavity/transmon stack with an on-chip spiral resonator controlled by a heavy-fluxonium ancilla
• Demonstrated measurement-free stabilization with logical-lifetime gain $1.59\pm 0.05$

Quasi-deterministic optical GKP generation — Yu et al. (2026):

• Combined near-deterministic photon-number-state injection, cat-state breeding, homodyne feedforward, and real-time displacement in a time-bin photonic processor
• Generated approximately 2,000 optical GKP states per second while operating at 250 kHz
• Demonstrated a one-dimensional approximate GKP state rather than a full two-dimensional fault-tolerant grid state; further squeezing and lower loss remain necessary for error correction

Key Metrics

Metric	Value	Notes	Fidelity reference
Pauli-averaged logical coherence gain	$2.27\pm 0.07$	Beyond-break-even superconducting GKP memory	Sivak et al. 2023
1Q logical process fidelity	0.960	Best reported trapped-ion GKP gate in the cited universal gate set	Matsos et al. 2025
2Q logical process fidelity	0.680	First demonstrated two-GKP-qubit CZ gate	Matsos et al. 2025
GKP Bell-state fidelity	0.842	Direct preparation from the two motional-mode ground states	Matsos et al. 2025
GKP qutrit / ququart coherence gain	$1.82\pm 0.03$ / $1.87\pm 0.03$	Beyond-break-even higher-dimensional GKP memories	Brock et al. 2025
Planar GKP logical-lifetime gain	$1.59\pm 0.05$	Heavy-fluxonium ancilla and on-chip spiral resonator; preprint	Jha et al. 2026
Quasi-deterministic optical GKP generation rate	$\sim 2,000$ states/s	250-kHz photonic experiment; demonstrated state was one-dimensional and not yet fault tolerant	Yu et al. 2026
Integrated-photonic symmetric effective squeezing	$0.62\pm 0.02$ dB	Generated GKP state remained below fault-tolerant quality	Larsen et al. 2025
Ideal logical-comb spacing	$2\sqrt{\pi }$	$q$-quadrature spacing within either logical basis state	Gottesman et al. 2001
Ideal correctable shift radius	$<\sqrt{\pi }/2$	For isolated square-lattice shift errors before modular ambiguity	Gottesman et al. 2001

Note: Gate fidelities here are logical-level process fidelities. Coherence gains compare an encoded memory with the best relevant physical component in the same experiment; they are not bare-device $T_1$ values.

References

Original proposal

• D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator,” Phys. Rev. A 64, 012310 (2001) — arXiv:quant-ph/0008040

Experimental demonstrations

• C. Flühmann et al., “Encoding a qubit in a trapped-ion mechanical oscillator,” Nature 566, 513 (2019) — arXiv:1807.01033
• P. Campagne-Ibarcq et al., “Quantum error correction of a qubit encoded in grid states of an oscillator,” Nature 584, 368 (2020) — arXiv:1907.12487
• V. V. Sivak et al., “Real-time quantum error correction beyond break-even,” Nature 616, 50 (2023) — arXiv:2211.09116
• V. G. Matsos et al., “Universal quantum gate set for Gottesman–Kitaev–Preskill logical qubits,” Nature Physics 21, 1664 (2025) — arXiv:2409.05455
• B. L. Brock et al., “Quantum error correction of qudits beyond break-even,” Nature 641, 612 (2025) — arXiv:2409.15065
• M. V. Larsen et al., “Integrated photonic source of Gottesman–Kitaev–Preskill qubits,” Nature 642, 587 (2025)
• S. Yu et al., “Extensible universal photonic quantum computing with nonlinearity,” Nature Photonics 20, 1166 (2026) — arXiv:2602.06544
• S. R. Jha et al., “Bosonic Error Correction with Fluxonium,” arXiv:2609.26880 (2026)

Linked Papers

• gottesman-2001-gkp
• campagneibarcq-2020-error-correction-qubit
• flhmann-2019-encoding-qubit-trapped
• michael-2016-binomial-codes
• sivak-2023-real-time-error
• matsos-2025-universal-gkp-logical-gates
• brock-2025-quantum-error-correction-qudits-beyond-break-even
• larsen-2025-integrated-photonic-gkp-source
• jha-2026-bosonic-error-correction-fluxonium
• yu-2026-extensible-universal-photonic-computing
• shiina-2026-high-rate-and-resource-efficient-all-photonic
• vyas-2026-homomorphic-aggregation-of-continuous-variable
• kumar-2026-deterministic-universal-logical-gates
• roberts-2026-bosonic-codes-from-compact
• wadewitz-2026-probabilistic-error-cancellation-for
• burgath-2026-spin-grid-states-for
• koudia-2026-joint-gkp-encoding-for
• aoki-2026-remote-gottesman-kitaev-preskill-entanglement

Evergreen context

• bosonic-code-hierarchy — the broader map of bosonic encodings
• dispersive-readout-mechanism — superconducting GKP experiments still cash out as repeated ancilla-assisted stabilizer readout inside a cavity-transmon stack
• motional-mode-coupling-in-ion-traps — the shared phonon-bus coupling underlying trapped-ion GKP implementations (Flühmann et al. 2019)
• threshold-theorem — why finite-energy GKP states matter for fault-tolerant thresholds
• erasure-error-vs-pauli-error — clarifies what kinds of oscillator errors GKP is designed to convert and correct

Related Entries

• cat-codes — alternative bosonic code using coherent state superpositions
• bosonic-qubit — umbrella entry for oscillator-encoded logical qubits
• circuit-qed — underlying hardware platform for superconducting GKP
• binomial-codes — another bosonic code family
• trapped-ion-qubit — motional-mode platform for GKP state preparation and logical gates
• continuous-variable-photonic-qubit — optical platform for integrated GKP resource states
• fluxonium — ancilla used in the 2026 planar superconducting GKP demonstration