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

TL;DR

This paper extends topological stabilizer codes to higher dimensions, enabling transversal non-Clifford gates and magic state preparation, surpassing previous bounds in fault-tolerant quantum computation.

Contribution

It introduces a broad class of $n$-dimensional Clifford hierarchy stabilizer codes with transversal non-Clifford gates, leveraging automorphism symmetries and code switching techniques.

Findings

First transversal non-Clifford gates in 2D stabilizer codes, including T and CS gates.

Transversal logical $rac{ ext{T}}{2}$ gate constructed in 3D non-Clifford stabilizer code.

Surpasses Bravyi-K"onig bound by achieving gates at the $(n+1)$-th level of Clifford hierarchy in $n$ dimensions.

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. Our constructions surpass the Bravyi-K\"onig bound by achieving the logical gates in the $(n+1)$-th level of Clifford hierarchy in $n$ spatial dimension.