Clifford-hierarchy stabilizer formalism with applications to twisted quantum doubles

Recent work has shown that quantum double models can be used to circumvent the Bravyi-Koenig bound. In particular, this is achieved through code-switching protocols in which non-Pauli stabilizer phases appear as intermediate codes. We introduce the X C(k) d stabilizer formalism, which generalizes the Pauli stabilizer formalism by combining Pauli- X operators with diagonal operators drawn from a fixed level k of the Clifford hierarchy on prime-dimensional qudits. This yields a natural class of beyond-Pauli stabilizer models, including many twisted quantum double phases. We classify a large class of these X C(k) d stabilizer states, showing that they are uniform superpositions over affine-linear subspaces with diagonal phases. In particular, when the stabilizers of a state require operators from level k of the Clifford hierarchy, we show that the diagonal gates required to produce the phases in the wavefunction sit one level higher. We further show that every 2D twisted quantum double D ω(Z p n) with ω a product of type-III cocycles and p an odd prime admits a realization using only Clifford (k=2) stabilizers, and we conjecture that phases where ω contains a type-I or type-II cocycle require level- p Clifford-hierarchy stabilizers for p-dimensional qudits. Finally, we demonstrate that a pair of qudit toric codes may be deterministically prepared in a logical magic state by measuring X C(k) d stabilizers. Together, these results provide a concrete stabilizer framework for topological codes beyond Pauli stabilizer models and their logical gates.