Clifford-hierarchy stabilizer formalism with applications to twisted quantum doubles
Abstract
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 X𝒞(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 X𝒞(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 cal Dω(ℤpn) 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 X𝒞(k)d stabilizers. Together, these results provide a concrete stabilizer framework for topological codes beyond Pauli stabilizer models and their logical gates.
Related
- Improved bounds on stabilizer extent and Clifford rank
- When is the Clifford hierarchy generalized semi-Clifford?
- A Cohomological Characterization of the Clifford Hierarchy
- Hierarchy of Rényi Coherent Information in Stabilizer Codes
- Semi-Cliffordness of the Clifford hierarchy for a single qudit in composite dimensions
BibTeX
@misc{fechisin2026cliffordhierarchy,
title = {{Clifford-hierarchy stabilizer formalism with applications to twisted quantum doubles}},
author = {Christopher Fechisin and Kieran Cooney and Mathi Raja and Victor V. Albert and Dominic J. Williamson and Kyle Kawagoe and Seth Musser},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00464},
eprint = {2610.00464},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00464},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Clifford-hierarchy stabilizer formalism with applications to twisted quantum doubles
AU - Christopher Fechisin
AU - Kieran Cooney
AU - Mathi Raja
AU - Victor V. Albert
AU - Dominic J. Williamson
AU - Kyle Kawagoe
AU - Seth Musser
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00464
UR - https://arxiv.org/abs/2610.00464
AB - 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 X𝒞^(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 X𝒞^(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 cal 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 X𝒞^(k)_d stabilizers. Together, these results provide a concrete stabilizer framework for topological codes beyond Pauli stabilizer models and their logical gates.
ER -