Spin GKP code
An analogue of the single-mode GKP code designed for atomic ensembles. Was designed by using the Holstein-Primakoff mapping  (see also ) to pull back the phase-space structure of a bosonic system to the compact phase space of a quantum spin. A different construction emerges depending on which particular expression for GKP codewords is pulled back.
Protect against errors native to spin systems like random rotations and stochastic relaxation.
Approximate Clifford-group generators are composed of Hamiltonians at most quadratic in angular momentum operators of two spin systems. Assuming that these generators can be implemented with high fidelity, a magic state can be prepared from an atomic ensemble analog of the vacuum state.
- Square-lattice GKP code — Spin-GKP code constructions utilize the Holstein-Primakoff mapping  (see also ) to convert various expressions for square-lattice GKP states into codes for spin systems.
- Landau-level spin code — The spin-GKP (Landau-level) code is a GKP-like encoding in (Landau-level) spin coherent states.
- S. Omanakuttan and T. J. Volkoff, “Spin squeezed GKP codes for quantum error correction in atomic ensembles”, (2023) arXiv:2211.05181
- T. Holstein and H. Primakoff, “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet”, Physical Review 58, 1098 (1940) DOI
- C. D. Cushen and R. L. Hudson, “A quantum-mechanical central limit theorem”, Journal of Applied Probability 8, 454 (1971) DOI
- D. W. Berry et al., “Simulating Hamiltonian Dynamics with a Truncated Taylor Series”, Physical Review Letters 114, (2015) arXiv:1412.4687 DOI
- G. H. Low and I. L. Chuang, “Hamiltonian Simulation by Qubitization”, Quantum 3, 163 (2019) arXiv:1610.06546 DOI
- Z. Holmes et al., “Quantum algorithms from fluctuation theorems: Thermal-state preparation”, Quantum 6, 825 (2022) arXiv:2203.08882 DOI
“Spin GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/spin_gkp