A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the Bravyi-König bound for $n$-dimensional topological stabilizer codes. In this Letter, we extend topological Pauli stabilizer codes to a broad class of $n$-dimensional Clifford hierarchy stabilizer codes. These codes correspond to the $(n+1)\mathrm{D}$ 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 $\sqrt{\mathrm{T}}$ 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.

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• Quantum error correction
• Quantum gates
• Quantum information theory
• Topological order