Classification of transversal gates in qubit stabilizer codes

Transversal operations in quantum error correction, that is logical gates that are executed by applying a set of gates in parallel, are the most straightforward form of fault-tolerant quantum logic. Naturally, characterizing the set of logical gates that can be implemented transversally for a class of codes is of great importance to fault-tolerant quantum architectures. In this work, we classify the set of logical gates that be realized by applying individual diagonal gates transversally. Namely, we show that for qubit stabilizer subspace codes, the set of logical gates that can be implemented by an operation of the form Z(θ)⊗n are restricted to have entries of the form e k along their diagonal, where c is an integer. We show that these results imply that logical gates implemented in this manner must belong to the Clifford hierarchy for all stabilizer codes and moreover the single and multi-qubit gates are restricted to be elements of the same level of the Clifford hierarchy.