Transversal Clifford-Hierarchy Gates via Non-Abelian Surface Codes

TL;DR

This paper introduces a method to implement transversal gates at any level of the Clifford hierarchy using non-Abelian surface codes in 2D, surpassing previous limitations and enabling fault-tolerant quantum computation.

Contribution

It constructs 2D transversal gates at arbitrary Clifford hierarchy levels using non-Abelian surface codes, extending the capabilities of quantum error correction.

Findings

Realized phase gates at any Clifford hierarchy level in 2D

Proposed a non-Abelian stabilizer formalism for dihedral groups

Achieved qubit-only implementation of high-level logical gates

Abstract

We present an entirely 2D transversal realization of phase gates at any level of the Clifford hierarchy, and beyond, using non-Abelian surface codes. Our construction encodes a logical qubit in the quantum double $D(G)$ of a non-Abelian group $G$ on a triangular spatial patch. The logical gate is implemented transversally by stacking on the spatial region a symmetry-protected topological (SPT) phase specified by a group 2-cocycle. The Bravyi--K\"onig theorem limits the unitary gates implementable by constant-depth quantum circuits on Pauli stabilizer codes in $D$ dimensions to the $D$-th level of the Clifford hierarchy. We bypass this limitation, by constructing transversal unitary gates at arbitrary levels of the Clifford hierarchy purely in 2D, without sacrificing locality or fault tolerance, at the cost of using the quantum double of a non-Abelian group $G$. Specifically, for $G = D_{4N}$, the dihedral group of order $8N$, we realize the phase gate $T^{1/N} = \mathrm{diag}(1, e^{i\pi/(4N)})$ in the logical $\overline{Z}$ basis. In this context we propose a non-abelian stabilizer group formalism, which we work out for dihedral groups. For $8N = 2^n$, the logical gate lies at the $n$-th level of the Clifford hierarchy and, importantly, has a qubit-only realization: we show that it can be constructed in terms of Clifford-hierarchy stabilizers for a code with $n$ physical qubits on each edge of the lattice. We also discuss code-switching to the $\mathbb{Z}_2 \times \mathbb{Z}_2$ and $\mathbb{Z}_2$ surface-codes, which can be utilized for the quantum error correction in this setup.