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arXiv:2610.00464

Clifford-hierarchy stabilizer formalism with applications to twisted quantum doubles

Christopher Fechisin, Kieran Cooney, Mathi Raja, Victor V. Albert, Dominic J. Williamson, Kyle Kawagoe, Seth Musser

quant-phcond-mat.str-elSubmitted Sep 30, 2026• Announced Oct 2, 2026• 0 citations
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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 $X\mathcal C^{(k)}_d$ 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 $X\mathcal C^{(k)}_d$ 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 ${\cal D}^ω(\mathbb{Z}_p^n)$ 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 $X\mathcal C^{(k)}_d$ stabilizers. Together, these results provide a concrete stabilizer framework for topological codes beyond Pauli stabilizer models and their logical gates.

Comments
19+14 pages, 4+3 figures

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The Clifford hierarchy C1⊂C2⊂⋯\mathsf{C}_1 \subset \mathsf{C}_2 \subset \cdotsC1​⊂C2​⊂⋯ was introduced by Gottesman and Chuang (arXiv:quant-ph/9908010) to characterize gates that admit fault-tolerant implementation by gate teleportation. Yet, despite its rich mathematical structure and the attention it has received in recent years, little is known about Ck\mathsf{C}_kCk​ for k>3k > 3k>3. Most progress has focused on identifying structural properties of restrictions of the hierarchy, such as diagonal gates and gates on systems of small dimension ddd or with few qudits. The generalized semi-Clifford conjecture, proposed by Zeng et al. (arXiv:0712.2084), states that every gate in ∪kCk\cup_k \mathsf{C}_k∪k​Ck​ is, up to multiplication by Cliffords, the product of a permutation and a diagonal matrix. Beigi and Shor proved the case d=2d=2d=2, k=3k=3k=3 (arXiv:0810.5108) and Pllaha et al. found an alternative proof by exploiting fixed points of the conjugation map induced by (a Clifford correction of) U∈C3U \in \mathsf{C}_3U∈C3​ on the span of maximal stabilizer subgroups (arXiv:2006.14040). By extending their fixed-point arguments to the group Γ1(U)Γ_1(U)Γ1​(U) generated by UPU†U \mathsf{P} U^\daggerUPU† and beyond, we prove the conjecture for k≤4k \leq 4k≤4 and any prime dimension ddd. Our proof centers on conjugation groups Γ1(U),Γ2(U),…Γ_1(U), Γ_2(U), \ldotsΓ1​(U),Γ2​(U),… of U∈CkU \in \mathsf{C}_kU∈Ck​, which we expect to be a useful tool in the study of the Clifford hierarchy more generally. We also show a natural sufficient condition on such groups for gates in higher levels to be generalized semi-Clifford.
We develop a simple and systematic route to higher levels of the qubit Clifford hierarchy by coherently controlling Clifford operations. Our approach is based on Pauli periodicity, defined for a Clifford unitary UUU as the smallest integer m≥1m\ge 1m≥1 such that U2mU^{2^{m}}U2m is a Pauli operator up to phase. We prove a sharp controlled-jump rule showing that the controlled gate CUCUCU lies strictly in level m+2m+2m+2 of the hierarchy, and equivalently that CUCUCU lies in level kkk if U2k−2U^{2^{k-2}}U2k−2 is Pauli while no smaller positive power of UUU is Pauli. We further quantify the resources required to realize large level jumps in the Clifford hierarchy by proving an essentially tight upper bound on Pauli periodicity as a function of the number of qubits, which implies that accessing high hierarchy levels through controlled Cliffords requires a number of target qubits that grows exponentially with the desired level. We complement this limitation with explicit infinite families of Pauli-periodic Cliffords whose controlled versions achieve asymptotically optimal jumps. As an application, we propose a protocol for preparing logical catalyst states that enable logical Z1/2kZ^{1/2^k}Z1/2k phase gates via phase kickback from a single jumped Clifford.