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arXiv:2511.02900 · Physical Review Letters

Clifford Hierarchy Stabilizer Codes: Transversal Non-Clifford Gates and Magic

Ryohei Kobayashi, Guanyu Zhu, Po-Shen Hsin

quant-phcond-mat.str-elhep-thSubmitted Nov 4, 2025• Announced Nov 6, 2025• 1 citation• Revised May 19, 2026 (v3)
v1 · Nov 4, 2025v2 · Dec 14, 2025v3 · May 19, 2026
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Abstract

A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-König bound for $n$-dimensional topological stabilizer codes. In this work, we extend topological Pauli stabilizer codes to a broad class of $n$-dimensional Clifford hierarchy stabilizer codes. These codes correspond to the $(n+1)$D Dijkgraaf-Witten gauge theories with non-Abelian topological order. We construct transversal non-Clifford gates through automorphism symmetries represented by cup products. In 2D, we obtain the first transversal non-Clifford logical gates including T and CS for Clifford stabilizer codes, using the automorphism of the twisted $\mathbb{Z}_2^3$ gauge theory (equivalent to $\mathbb{D}_4$ topological order). We also combine it with the just-in-time decoder to fault-tolerantly prepare the logical T magic state in $O(d)$ rounds via code switching. In 3D, we construct a transversal logical $\sqrt{\text{T}}$ gate in a non-Clifford stabilizer code at the third level of the Clifford hierarchy, located on a tetrahedron corresponding to a twisted $\mathbb{Z}_2^4$ gauge theory. Our constructions surpass the Bravyi-König bound by achieving the logical gates in the $(n+1)$-th level of Clifford hierarchy in $n$ spatial dimension.

Journal reference
Phys. Rev. Lett. 136, 250802 (2026)
DOI
10.1103/j2q4-bnc6
Comments
5+17 pages, 11 figures. Close to the published version

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Gauge theories are important descriptions for many physical phenomena and systems in quantum computation. Automorphism of gauge group naturally gives global symmetries of gauge theories. In this work we study such symmetries in gauge theories induced by automorphisms of the gauge group, when the gauge theories have nontrivial topological actions in different spacetime dimensions. We discover the automorphism symmetry can be extended, become a higher group symmetry, and/or become a non-invertible symmetry. We illustrate the discussion with various models in field theory and on the lattice. In particular, we use automorphism symmetry to construct new transversal non-Clifford logical gates in topological quantum codes. In particular, we show that 2+1d ZN\mathbb{Z}_NZN​ qudit Clifford stabilizer models can implement non-Clifford transversal logical gate in the 4th level ZN\mathbb{Z}_NZN​ qudit Clifford hierarchy for N≥3N\geq 3N≥3, extending the generalized Bravyi-König bound proposed in the companion paper [arXiv:2511.02900] for qubits.
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