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:
Gross spin code.
Cousins:
Qubit 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:
Spin code.

## References

- [1]
- J. A. Gross, “Designing Codes around Interactions: The Case of a Spin”, Physical Review Letters 127, (2021). DOI; 2005.10910