## Description

A spin code designed to realize a discrete group of gates using \(SU(2)\) rotations, which are realized transversally if the single spin is treated as a collective spin of several spin-half subsystems. Codewords are subspaces of a spin's Hilbert space that house irreducible representations (irreps) of a discrete subgroup of \(SU(2)\).

The first realization [1] used the single-qubit Clifford group (i.e., the binary octahedral (\(2O\)) subgroup of \(SU(2)\)). Code construction 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.

A simple example of a codespace is a projection onto an instance of a particular irrep of \(2O\), referred to as either \( \varrho_4 \) or \( \varrho_5 \). In the case of only one instance of the desired irrep present in the spin, the projection is created as follows: \begin{align} P_\varrho = \frac{\text{dim} \varrho}{|2O|} \sum_{g \in 2O} \chi_\varrho (g)^* D(g)~, \tag*{(1)}\end{align} where \(D(g)\) is the \(SU(2)\) Wigner matrix corresponding to group element \(g\), and the character \(\chi_\varrho (g) = \text{tr}(\varrho(g))\) is the trace of the desired irrep evaluated at a group element.

Logical Pauli matrices \(\overline{\sigma}_w\) are defined using the above projection and the angular momentum operators: \begin{align} \overline{\sigma}_w = i P_\varrho e^{-i \pi J_w} P_\varrho~. \tag*{(2)}\end{align} Finally, \(|\overline{0} \rangle\) is defined as the \(+1\) eigenvalue of \(\overline{\sigma}_z\) and \(|\overline{1} \rangle = \overline{\sigma}_x |\overline{0} \rangle \).

## Transversal Gates

## Gates

## Parent

## Child

- \(((7,2,3))\) permutation-invariant code — The \(((7,2,3))\) permutation-invariant code can be interpreted as a spin-\(7/2\) Clifford code [1].

## Cousin

- Qubit code — Certain Clifford codes yield qubit codes with non-trivial distance when the single spin is treated as a collective spin of several qubits.

## References

- [1]
- J. A. Gross, “Designing Codes around Interactions: The Case of a Spin”, Physical Review Letters 127, (2021) arXiv:2005.10910 DOI
- [2]
- S. Omanakuttan and J. A. Gross, “Multispin Clifford codes for angular momentum errors in spin systems”, (2023) arXiv:2304.08611

## Page edit log

- Victor V. Albert (2022-05-25) — most recent
- Thomas Wrona (2022-05-18)

## Cite as:

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