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.

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 \(|\overline{1} \rangle = \overline{\sigma}_x |\overline{0} \rangle \).