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

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}₂³ 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{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}₂⁴ gauge theory. Due to the potential single-shot code-switching properties of these codes, one could achieve the 4th level of Clifford hierarchy with an O\(d³\) space-time overhead, avoiding the tradeoff observed in 2D. We propose a conjecture extending the Bravyi-König bound to Clifford hierarchy stabilizer codes, with our explicit constructions surpassing the Bravyi-König bound for achieving the logical gates in the (n+1)-th level of Clifford hierarchy in n spatial dimension.