OAT-GKP Codes in Collective Spin Systems

• OAT-GKP codes are a quantum error correction paradigm that encodes qubits in spin ensembles using discretely rotated and spin-squeezed states on the Bloch sphere.
• They leverage a one-axis twisting Hamiltonian to generate comb-like lattice structures with well-defined stabilizers in the compact SU(2) phase space.
• This framework supports fault-tolerant gate operations and robust error diagnosis under noise models specific to atomic ensembles.

OAT-GKP codes extend the Gottesman-Kitaev-Preskill (GKP) paradigm for quantum error correction into the setting of collective spin systems, leveraging the one-axis twisting (OAT) interaction to implement a “comb” code structure on the Bloch sphere. These codes encode qubits in spin ensembles through superpositions of discretely rotated, spin-squeezed states, producing analogues of continuous-variable GKP codes within the compact phase space of SU(2). The OAT-GKP framework supports efficient error diagnosis and correction under several important noise models specific to atomic ensembles and inherits key structural and operational features from bosonic GKP codes. OAT-GKP codes are distinguished from related spin-based codes by their lattice structure, error-correcting performance, and realizable approximately fault-tolerant gate set, rooted in the rich translation between spin and bosonic encodings (Omanakuttan et al., 2022).

1. Code Construction

The OAT-GKP code is formulated for an ensemble of $N$ spin-1/2 particles (total spin $J=N/2$) within the symmetric subspace. The central construction relies on the one-axis twisting (OAT) Hamiltonian,

$H_{\mathrm{OAT}} = \chi S_z^2$

and its associated unitary evolution,

$U_{\mathrm{OAT}}(\tau) = \exp(-i\chi\tau S_z^2).$

Preparing a coherent spin state (CSS) $|J, J_z=J\rangle$ and evolving under $U_{\mathrm{OAT}}(\tau)$ yields spin squeezing along the $z$-axis, deforming the state on the Bloch sphere. Subsequent discrete rotations by

$\Delta\theta = \sqrt{2\pi/N}$

about the $y$-axis generate a family of states

$$|\psi_t\rangle = \exp(-it\Delta\theta S_y) U_{\mathrm{OAT}}(\tau)|J,J\rangle, \quad t\in\{-T, ..., +T\}$$

localized at evenly spaced longitudes. Superpositions of these states, with appropriately chosen phase and amplitude weights, form the logical codewords. A uniform-lattice OAT-GKP codeword is defined as

$J=N/2$0

with $J=N/2$1 controlling the degree of squeezing and $J=N/2$2 as the initial CSS. These codewords can be re-expressed in the Dicke basis as weighted sums with analytically-known Wigner-$J=N/2$3 coefficients. Gaussian-enveloped structures are achievable by modulating the amplitude weights via combinatorial factors, effectively tuning the peak sharpness and lattice envelope.

2. Stabilizers and Lattice Structure

OAT-GKP codewords are eigenstates of two commuting lattice-translation “stabilizer” operators, which mirror the role of stabilizers in square-lattice CV GKP codes but are embedded in SU(2):

$J=N/2$4

These operators implement minimal shifts along the $J=N/2$5 and $J=N/2$6 directions. The simultaneous $J=N/2$7 eigenspace corresponds to the logical code space, with repeated shifts generating an approximately $J=N/2$8 lattice grid over the compact phase space. This structure generates comb-like features in the $J=N/2$9 and $H_{\mathrm{OAT}} = \chi S_z^2$0 eigenbases, analogous to the rectilinear lattice of GKP states in the $H_{\mathrm{OAT}} = \chi S_z^2$1 and $H_{\mathrm{OAT}} = \chi S_z^2$2 quadratures of a bosonic oscillator (Omanakuttan et al., 2022).

3. Finite-$H_{\mathrm{OAT}} = \chi S_z^2$3 Analysis and Central Limit Connections

The OAT-GKP construction is rigorously connected to the standard CV GKP code via a quantum central limit theorem (QCLT). An isometric embedding $H_{\mathrm{OAT}} = \chi S_z^2$4 from the symmetric spin subspace to Fock space satisfies

$H_{\mathrm{OAT}} = \chi S_z^2$5

where $H_{\mathrm{OAT}} = \chi S_z^2$6 is the annihilation operator. In the $H_{\mathrm{OAT}} = \chi S_z^2$7 limit, evolutions and lattice structures (e.g., the OAT squeezing $H_{\mathrm{OAT}} = \chi S_z^2$8) map onto their CV counterparts (e.g., Kerr interaction for GKP codes). Thus, OAT-GKP codes are “pullbacks” of CV GKP combs onto SU(2); for large $H_{\mathrm{OAT}} = \chi S_z^2$9, they reproduce the CV logical comb structure and error-correcting properties.

For finite $U_{\mathrm{OAT}}(\tau) = \exp(-i\chi\tau S_z^2).$0, the width of peaks in the $U_{\mathrm{OAT}}(\tau) = \exp(-i\chi\tau S_z^2).$1-basis,

$U_{\mathrm{OAT}}(\tau) = \exp(-i\chi\tau S_z^2).$2

determines the resolvability of adjacent lattice sites. The code’s efficacy depends on maintaining $U_{\mathrm{OAT}}(\tau) = \exp(-i\chi\tau S_z^2).$3, i.e., well-squeezed, separated peaks.

4. Error Correction and Fidelity Under Noise

OAT-GKP codes address two primary noise models relevant to atomic-ensemble systems:

• Stochastic relaxation (spin-analogue of photon loss): Each spin can “jump” from $U_{\mathrm{OAT}}(\tau) = \exp(-i\chi\tau S_z^2).$4 to $U_{\mathrm{OAT}}(\tau) = \exp(-i\chi\tau S_z^2).$5 independently, governed by channel

$U_{\mathrm{OAT}}(\tau) = \exp(-i\chi\tau S_z^2).$6

• Isotropic ballistic dephasing: SU(2)-invariant random rotations of all spins,

$U_{\mathrm{OAT}}(\tau) = \exp(-i\chi\tau S_z^2).$7

Channel fidelity $U_{\mathrm{OAT}}(\tau) = \exp(-i\chi\tau S_z^2).$8 for an optimally recovered encoded qubit is computed via a reduced SDP (size $U_{\mathrm{OAT}}(\tau) = \exp(-i\chi\tau S_z^2).$9). Recovery exploits the distinguishability and redundancy of the spin GKP comb after noise; optimized recovery “reassembles” the logical state. Numerically, for $|J, J_z=J\rangle$0, OAT-GKP codes outperform binomial and cat codes under both stochastic relaxation ($|J, J_z=J\rangle$1) and isotropic dephasing ($|J, J_z=J\rangle$2 for $|J, J_z=J\rangle$3), provided the comb structure is not destroyed (Omanakuttan et al., 2022).

5. Fault-Tolerant Gate Set Implementation

OAT-GKP codes inherit an approximately fault-tolerant universal gate set by translating canonical CV GKP operations via the QCLT correspondence:

• SUM (CV CNOT): $|J, J_z=J\rangle$4
• CV Fourier ($|J, J_z=J\rangle$5): spin Fourier $|J, J_z=J\rangle$6
• CV phase gate $|J, J_z=J\rangle$7: spin $|J, J_z=J\rangle$8

All these unitaries preserve the OAT-GKP code up to correctable errors. Magic-state injection is performed via ancilla, exploiting two SUM operations to prepare $|J, J_z=J\rangle$9-type logical states. Error propagation through such circuits remains within the recovery radius set by the stabilizer lattice, ensuring resilience under concatenated noise-agnostic GKP circuits (Omanakuttan et al., 2022).

6. Comparison with Other GKP-Based and Spin Codes

OAT-GKP codes generalize and extend the CV GKP code structure to the collective-spin domain, offering a direct spin-based realization that retains the core features of comb encoding for quantum error correction. Compared to alternative spin codes, such as two-axis countertwisting (TACT) GKP, cat codes, or binomial codes, OAT-GKP shows competitive or superior channel fidelity under experimentally relevant atomic-ensemble noise. The TACT-based spin GKP code slightly outperforms OAT-GKP under stochastic relaxation, but both far exceed non-comb codes in error resilience for moderate system sizes ($U_{\mathrm{OAT}}(\tau)$0). Their lattice-based structure and optimal-recovery performance position OAT-GKP as a viable tool for robust quantum information processing and highlight the efficacy of GKP-inspired approaches in platforms beyond bosonic CV systems (Omanakuttan et al., 2022).

7. Outlook and Implications

OAT-GKP codes unify the conceptual frameworks of bosonic and spin-based error correction by exploiting lattice structures in compact phase space. This approach provides a blueprint for robust, hardware-efficient quantum information encoding in atomic ensembles and possibly other finite-dimensional systems. By leveraging the quantum central limit connection, OAT-GKP codes are expected to facilitate scalable, approximately fault-tolerant quantum computing with operational gate sets directly inherited from their CV analogues. The diversity of codeword lattice shapes, error recovery methods, and physical implementations enabled by this construction continues to motivate theoretical and experimental investigations into high-performance quantum error correction outside the standard bosonic domain (Omanakuttan et al., 2022).