Constant-Depth Clifford-Hierarchy Gates via Non-Abelian Surface Codes

We present an entirely 2D constant-depth realization of topologically protected 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 by a constant-depth circuit constructed from stacking on the spatial region a symmetry-protected topological (SPT) phase specified by a group 2-cocycle and boundary counter-terms. 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 constant-depth 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 double surface-code $D(\mathbb{Z}_2\times\mathbb{Z}_2)$, to complete a universal gate-set in this setup.