General Gate Teleportation and the Inner Structure of Its Clifford Hierarchies

The quantum gate teleportation mechanism allows for the fault‐tolerant implementation of “Clifford hierarchies” of gates assuming, among other things, a fault‐tolerant implementation of the Pauli gates. We discuss how this method can be extended to assume the fault‐tolerant implementation of any orthogonal unitary basis of operators, in such a way that each of these bases can be used to implement its own hierarchy of gates. Noticing that all of these hierarchies are defined through the same algebraic construction, we study this construction from a group‐theoretic perspective. This allows us to deduce general results concerning all the hierarchies. We also discuss alternative hierarchies, study their properties, and compute some of their first few levels.