## Description

A single-spin code designed to realize a discrete group of gates using \(SU(2)\) rotations. 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 (effectively, the binary octahedral, or \(2O\) 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 matrix 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 \(|\overline{1} \rangle = \overline{\sigma}_x |\overline{0} \rangle \).

## Gates

## Parents

- Single-spin code
- Group-representation code — Clifford spin codes are group-representation codes with \(G\) being a subgroup of \(SU(2)\) [3].
- PI qubit code — Clifford codes for spins housing representations of \(SU(2)\) yield PI qubit codes with non-trivial distance when the single spin-\(n/2\) is treated as the permutationally invariant subspace of \(n\) qubits via the Dicke-state mapping. The subgroup of gates of a Clifford spin code is implemented transversally via this mapping [2].

## Children

- Binary dihedral PI code — Binary dihedral PI codes can be interpreted as Clifford single-spin codes.
- \(((7,2,3))\) Pollatsek-Ruskai code — The \(((7,2,3))\) Pollatsek-Ruskai 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, “Quantum error-correcting codes with a covariant encoding”, (2024) arXiv:2306.11621

## Page edit log

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

## Cite as:

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