CLASSIFICATION OF TRANSVERSAL GATES IN QUBIT STABILIZER CODES

This work classifies the set of diagonal gates that can implement a single or two-qubit transversal logical gate for qubit stabilizer codes. We show that individual physical diagonal gates on the underlying qubits that compose the code are restricted to have entries of the form e(i pi c/2k) along their diagonal, resulting in a similarly restricted class of logical gates that can be implemented in this manner. As such, we show that all diagonal logical gates that can be implemented transversally by individual physical diagonal gates must belong to the Clifford hierarchy. Moreover, we show that for a given stabilizer code, the two-qubit diagonal transversal gates must belting to the same level of Clifford hierarchy as the single-qubit diagonal transversal gates available for the given code. We use this result to prove a conjecture about arbitrary wersal gates made by Zeng et al. in 2007.