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

Abstract: A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-K\"onig 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. Due to the potential single-shot code-switching properties of these codes, one could achieve the 4th level of Clifford hierarchy with an $O(d^3)$ space-time overhead, avoiding the tradeoff observed in 2D. We propose a conjecture extending the Bravyi-K\"onig bound to Clifford hierarchy stabilizer codes, with our explicit constructions providing an upper bound of spatial dimension $(N-1)$ for achieving the logical gates in the $N^\text{th}$-level of Clifford hierarchy.