Spin GKP code[1] 


An analogue of the single-mode GKP code designed for atomic ensembles. Was designed by using the Holstein-Primakoff mapping [2] (see also [3]) 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.


Linear combination of unitaries method [46], which may be applicable to more general codewords.


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 [2] (see also [3]) 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
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Zoo Code ID: spin_gkp

Cite as:
“Spin GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/spin_gkp
@incollection{eczoo_spin_gkp, title={Spin GKP code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/spin_gkp} }
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Cite as:

“Spin GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/spin_gkp

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/spins/single_spin/spin_gkp.yml.