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

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)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 Z_{2}^{3} gauge theory (equivalent to 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[T] gate in a non-Clifford stabilizer code at the third level of the Clifford hierarchy, located on a tetrahedron corresponding to a twisted 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.