## 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. In cases where multiple copies of the irrep are present, one can try to optimize the distance of the code inside the multiplicity space.

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

## Parents

- Single-spin code
- Group-representation code — Clifford codes are group-representation codes with \(G\) being a subgroup of \(SU(2)\) [3].

## Children

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

## 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”, Physical Review A 108, (2023) arXiv:2304.08611 DOI
- [3]
- A. Denys and A. Leverrier, “Multimode bosonic cat codes with an easily implementable universal gate set”, (2023) arXiv:2306.11621

## 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