Transversal Clifford-Hierarchy Gates via Non-Abelian Surface Codes

Alison Warman; Sakura Schafer-Nameki

Paper

Transversal Clifford-Hierarchy Gates via Non-Abelian Surface Codes

Abstract

We present a purely 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

of a non-Abelian group

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--König theorem limits the unitary gates implementable by constant-depth quantum circuits on Pauli stabilizer codes in

dimensions to the

-th level of the Clifford hierarchy. We bypass this, by constructing transversal unitary gates at arbitrary levels of the Clifford hierarchy purely in 2D, without sacrificing locality or fault tolerance, however at the cost of using the quantum double of a non-Abelian group

. Specifically, for

, the dihedral group of order

, we realize the phase gate

in the logical

basis. For

, this gate lies at the

-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

physical qubits on each edge of the lattice. We also discuss code-switching to the

and

toric codes, which can be utilized for the quantum error correction in this setup.