Spin-GKP Codes: Quantum Error Correction in Spin Systems

• Spin-GKP codes are quantum error-correcting codes that adapt continuous-variable lattice structures to finite-dimensional collective spin systems, enabling robust qubit encoding.
• They leverage the quantum central limit theorem and Holstein–Primakoff approximation to transfer GKP grid structures to the compact SU(2) phase space, effectively mitigating both collective and local decoherence.
• Implementation schemes utilize collective operations such as two-axis counter-twisting and measurement-free local error recovery, paving the way for fault-tolerant universal quantum computation.

Spin-GKP codes are quantum error-correcting codes that map the lattice-based structure of Gottesman–Kitaev–Preskill (GKP) continuous-variable (CV) codes onto large collective spin systems, enabling robust quantum information storage and manipulation in atomic ensembles and related platforms. By leveraging the quantum central limit theorem (QCLT) and Holstein–Primakoff approximation, these codes pull back the phase-space grid structure from a bosonic mode to the finite-dimensional, compact SU(2) phase space of collective angular momentum. Spin-GKP codes demonstrate high error tolerance to both collective and local decoherence channels and provide a direct pathway to fault-tolerant universal quantum computation using only collective interactions and without requiring individual spin addressability (Omanakuttan et al., 2022, Omanakuttan et al., 24 Jan 2026).

1. Theoretical Foundations and Motivation

Continuous-variable GKP codes encode logical qubits into displaced lattice combs in the phase space of a single bosonic mode, with stabilizers $\hat S_q = e^{2i\sqrt{\pi}\hat p}$ and $\hat S_p = e^{-2i\sqrt{\pi}\hat q}$. The ideal codewords are eigenstates supported on an infinite grid in position or momentum space, e.g.,

$|0_L\> = \sum_{n\in\mathbb Z}|q = n\sqrt{2\pi}\>,$

which yield perfect correctability for shifts in $q$ or $p$ up to $\sqrt{\pi/2}$. Finite-energy variants ("finite-squeezing GKP") utilize Gaussian envelopes and are exceptionally robust against displacement, photon loss, and even certain non-Gaussian errors.

Spin-GKP codes adapt the GKP paradigm to the setting of $N$-particle symmetric spin ensembles (collective angular momentum $J=N/2$), where the natural state space corresponds to the symmetric subspace of $N$ two-level systems. Spin platforms—including atomic ensembles, trapped ions, and spinor Bose–Einstein condensates—are physically constrained to collective operations and experience both collective and local decoherence. The QCLT and Holstein–Primakoff mapping between spin operators and bosonic quadratures

$$V e^{\frac{\alpha}{\sqrt N}J_- - h.c.}|N,0\> \to D(\alpha)|0\>,$$

allow transfer of the GKP code structure to spin systems, thereby enabling robust quantum error correction in atomic systems that lack a direct bosonic mode (Omanakuttan et al., 2022, Omanakuttan et al., 24 Jan 2026).

2. Construction of Spin-GKP Codes

Spin-GKP codewords are realized as superpositions of spin coherent states (SCSs) within the symmetric subspace of $N$ spins, usually starting from the highest-weight state $|J, J_z=J\rangle$. Typical codeword families include:

• TACT-GKP Codes: Utilizing two-axis counter-twisting (TACT) squeezing and rotations (Hamiltonian $H_{TACT} = \chi(S_x^2 - S_y^2)$), the codewords take the form

$$\Big| \mu_{\lambda,z} \Big\rangle \propto \sum_{t=-T}^{T} e^{-2\pi\lambda^2 (t+\mu/2)^2} R_y\left(\sqrt{2\pi}t + \mu\sqrt{\frac{\pi}{2}}\right) e^{\frac{z}{2N}(J_{+}^2 - J_{-}^2)} |N,0\rangle,$$

with $\mu \in \{0,1\}$ labelling logical 0 and 1, and $T$ truncating the spin-lattice comb.

• SU(2)-Coherent GKP Codes ("spingkp", "unigkp"): Generalize the lattice to two dimensions on the Bloch sphere,

$$\big|\mu_\delta \big\rangle \propto \sum_{t_1, t_2 = -T}^T e^{- \frac{\pi}{2} \delta^2 \left( (2t_1+\mu)^2 + t_2^2 \right)} R_y(...) R_x(...) |N,0\rangle,$$

mimicking the CV code's square lattice in the spin setting.

• OAT-GKP Codes: Based on one-axis twisting (OAT) squeezing ($H_{OAT} = \chi J_z^2$), these provide spin-only comb codes with no known CV analogue but similar error-protecting structure.

Logical operators are mapped to collective spin rotations: $$X_L = e^{-i \sqrt{2\pi/N} J_y},\qquad Z_L = e^{-i \sqrt{2\pi/N} J_x}$$ which, in the $N\to\infty$ limit, reproduce the qubit algebra.

The Holstein–Primakoff mapping provides an explicit dictionary from bosonic GKP codes to permutation symmetric spin codes, enabling generalization to other bosonic codes (binomial, cat, etc.) (Omanakuttan et al., 24 Jan 2026).

3. Noise Models and Error Correction Performance

Spin-GKP codes are evaluated under physical noise models directly relevant to atomic ensembles:

• Stochastic Relaxation (Amplitude Damping Analogue): Each spin relaxes from $|1\rangle \to |0\rangle$ at a rate $\lambda$, analogously to photon loss. The channel acts as

$$T_t(\rho) = \sum_{\ell=0}^N \frac{(1-e^{-\lambda t})^\ell}{\ell!} e^{-\frac{\lambda t}{4}(N-2J_z)} (\mathcal S^{(1)})^\ell(\rho) e^{-\frac{\lambda t}{4}(N-2J_z)}.$$

• Isotropic Ballistic Dephasing: Fluctuating global fields yield uncertain collective rotations, described as a mixture over all rotation axes.
• Local Spin-Depolarizing Noise: Each spin is independently depolarized, modeled by

$$\frac{d\rho}{dt} = \frac{\gamma}{4} \sum_{n,i} [\sigma_i^{(n)} \rho \sigma_i^{(n)} - \rho].$$

Population leaks into SU(2) irreps with $J=N/2-1, N/2-2,...$, but the code's self-similarity across irreps (in the Holstein–Primakoff limit) preserves logical structure up to $\mathcal{O}(1/N)$ corrections.

The optimal entanglement fidelity is computed via a diversity combining approach, reducing the semi-definite program to manageable size by recasting encoding+noise and recovery as qubit $\leftrightarrow (N+1)$ maps. Numerical benchmarks for $N=64$ show that tactgkp, spingkp, and unigkp codes outperform binomial, "single-rail," and cat codes under collective relaxation and dephasing for relevant parameter regimes (Omanakuttan et al., 2022). Under local depolarization, the fidelity loss is dominated by leakage into neighboring irreps (rate $3\gamma/4$), and logical errors scale as $\mathcal{O}(1/N)$ (Omanakuttan et al., 24 Jan 2026).

4. Implementation Schemes

Preparation of spin-GKP states can be achieved via the linear combination of unitaries (LCU) protocol:

• Prepare an ancilla in a superposition $\sum_t \alpha_t |t\rangle_{anc}$.
• Apply controlled-unitary $$U_{select} = \sum_t |t\rangle\langle t| \otimes U(t)$$ targeting the data ensemble.
• Uncompute the ancilla and post-select on $|0\rangle_{anc}$.

For tactgkp(uni) codes, $U_{select}$ involves controlled rotations or two-axis counter-twisting operations, with an ancilla of dimension $2T+1$. The circuit is resource-efficient and robust to coherent errors on the ancilla (Omanakuttan et al., 2022).

Measurement-free local error recovery (MFLER) is enabled by the Holstein–Primakoff structure: Population that leaks into non-symmetric irreps due to local noise carries a self-similar code, which can be coherently swapped back into the symmetric subspace using two collective CNOT gates and an ancilla ensemble. This process requires no intermediate measurement or feed-forward (Omanakuttan et al., 24 Jan 2026).

5. Fault-Tolerant Logical Gate Set

Spin-GKP codes admit a universal logical gate set inherited from the CV GKP paradigm:

• SUM Gate (CNOT Analogue): $$\mathrm{SUM} = \exp\left(-2i \frac{J_x \otimes J_y}{N}\right)$$.
• Fourier (Hadamard) and Phase: $F = \exp(-i\pi/2 J_z)$ and $P = \exp(i J_x^2 / N)$.
• Pauli Operators: $X_L \approx e^{-2i\sqrt{2\pi/N} J_x}$ and $Z_L \approx e^{+2i\sqrt{2\pi/N} J_y}$.

Error propagation under these gates is tractable: Conjugation by $F$ swaps $J_x \leftrightarrow J_y$, and SUM spreads single-ensemble errors onto a second ensemble as correlated rotations, which remain correctable with subsequent noise-agnostic recovery. Non-Clifford resources (magic states) for universality can be distilled using GKP error correction applied to a spin coherent state via SUM (Omanakuttan et al., 2022, Omanakuttan et al., 24 Jan 2026).

6. Comparative Analysis and Robustness

The spin-GKP construction provides several key benefits:

Attribute	Spin-GKP Codes	Binomial/Cat Codes
Error Correction Scope	Collective + Local Noise	Typically limited to local errors
Physical Implementation	Collective controls in symmetric subspace	Often needs local addressability
Channel Fidelity (collective noise)	Superior for realistic atomic channels	Lower for stochastic relaxation
Fault-Tolerant Recovery	Diversity combining, MFLER protocols	Stabilizer syndrome/measurements

Spin-GKP codes correct any collective rotation/dissipation to first order if the Knill–Laflamme conditions are met in the symmetric subspace. Permutation invariance directly suppresses local decoherence. Numerical and tensor-network simulations verify robustness against both symmetric and inhomogeneous noise, with logical error rates decaying exponentially in the effective squeezing and code size (Omanakuttan et al., 2022, Omanakuttan et al., 24 Jan 2026).

7. Extensions, Challenges, and Outlook

Significant challenges include state preparation overheads (ancilla dimension, controlled rotations), all-to-all interaction requirements for SUM gates, and finite-$N$ corrections limiting fidelity to the ideal GKP behavior.

Potential extensions include: codes built from higher SU($d$) coherent states (multilevel atoms), concatenation with surface or LDPC codes to achieve full fault-tolerance thresholds, and experimental realization in platforms such as trapped ions (via Mølmer–Sørensen interactions), Rydberg atoms, and cold atoms in optical cavities. Optimization of spin-GKP lattice geometry on the Bloch sphere, invoking spherical code and kissing-number problems, is anticipated to yield enhanced performance for fixed system size (Omanakuttan et al., 2022).

A notable recent development is the general framework of Holstein–Primakoff spin codes, which applies to generic bosonic codes in permutation symmetric spin ensembles and introduces measurement-free local error recovery, expanding the robustness of spin-GKP codes to local and collective noise without local measurement or feed-forward (Omanakuttan et al., 24 Jan 2026).

Together, these advancements situate spin-GKP codes as a central architecture for practical, robust quantum error correction in spin-based hardware with only collective control resources.