Classification of transversal gates in qubit stabilizer codes

Understand

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 gates on the underlying qubits that compose the code are restricted to have entries of the form $e^{i \pi c/2^k}$ along their diagonal, resulting in a similarly restricted class of logical gates that can be implemented in this manner.
• Moreover, we show that all diagonal logical gates that can be implemented transversally by individual physical diagonal gates must belong to the Clifford hierarchy.
• Furthermore, we can use this result to prove a conjecture about transversal gates made by Zeng et al.