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.
Parents:
Finite-dimensional quantum error-correcting code.
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.

Gross spin code[1]
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.

Spin GKP code[2]
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