Welcome to the Spin Kingdom.

Spin code Encodes $$K$$-dimensional Hilbert space into a $$q^n$$-dimensional ($$n$$-qudit) Hilbert space, where the canonical qudit basis consists of states of a quantum mechanical spin. In other words, canonical single-qudit states $$|^\ell_m\rangle$$ are labeled by total angular momentum $$\ell$$ (either integer or half-integer) and its $$z$$-axis projection $$m$$, with $$q=2\ell+1$$. Protection: Spin codes are often designed to protect against $$SU(2)$$ rotations by small angles. Parent of: Single-spin code. Cousins: Qubit code.
Single-spin code Encodes $$K$$-dimensional Hilbert space into a $$2\ell+1$$-dimensional Hilbert space, where the latter is thought of as a spin-$$\ell$$ quantum system. This spin system can in turn be thought of as the maximally symmetric subspace or collective spin of $$2\ell$$ spin-half systems. Parents: Spin code. Parent of: Gross spin code, Spin GKP code.
A spin code designed to realize Clifford gates using $$SU(2)$$ rotations. Codewords are subspaces of a spin's Hilbert space that house irreducible representations (irreps) of the single-qubit Clifford group (i.e., the binary octahedral ($$2O$$) subgroup of $$SU(2)$$). This is done by restricting the $$SU(2)$$ irrep to $$2O$$, and determining the carrier spaces of any nontrivial irreps of $$2O$$. Since irreps of $$2O$$ do not appear in integer spins, half-integer spins are used. Parents: Single-spin code.
An analogue of the single-mode GKP code designed for atomic ensembles. Was designed by using the Holstein-Primakoff mapping [3] (see also [4]) 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. Parents: Single-spin code. Cousins: Gottesman-Kitaev-Preskill (GKP) code.

## References

[1]
J. A. Gross, “Designing Codes around Interactions: The Case of a Spin”, Physical Review Letters 127, (2021). DOI; 2005.10910
[2]
Sivaprasad Omanakuttan and T. J. Volkoff, “Spin squeezed GKP codes for quantum error correction in atomic ensembles”. 2211.05181
[3]
T. Holstein and H. Primakoff, “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet”, Physical Review 58, 1098 (1940). DOI
[4]
C. D. Cushen and R. L. Hudson, “A quantum-mechanical central limit theorem”, Journal of Applied Probability 8, 454 (1971). DOI