# Spin GKP code[1]

## Description

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.

## Protection

Protect against errors native to spin systems like random rotations and stochastic relaxation.

## Encoding

Linear combination of unitaries method [4][5][6], which may be applicable to more general codewords.

## Gates

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.

## Parent

## Cousin

- 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.

## References

- [1]
- S. Omanakuttan and T. J. Volkoff, “Spin squeezed GKP codes for quantum error correction in atomic ensembles”, (2022) arXiv:2211.05181
- [2]
- T. Holstein and H. Primakoff, “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet”, Physical Review 58, 1098 (1940) DOI
- [3]
- C. D. Cushen and R. L. Hudson, “A quantum-mechanical central limit theorem”, Journal of Applied Probability 8, 454 (1971) DOI
- [4]
- D. W. Berry et al., “Simulating Hamiltonian Dynamics with a Truncated Taylor Series”, Physical Review Letters 114, (2015) arXiv:1412.4687 DOI
- [5]
- G. H. Low and I. L. Chuang, “Hamiltonian Simulation by Qubitization”, Quantum 3, 163 (2019) arXiv:1610.06546 DOI
- [6]
- Z. Holmes et al., “Quantum algorithms from fluctuation theorems: Thermal-state preparation”, Quantum 6, 825 (2022) arXiv:2203.08882 DOI

## Page edit log

- Victor V. Albert (2022-11-13) — most recent
- Sivaprasad Omanakuttan (2022-11-13)

## 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/spin_gkp.yml.