Interaction cost of nonlocal gates

We introduce the interaction cost of a nonlocal gate as the minimal time of interaction required to perform the gate when assisting the process with fast local unitaries. This cost, of interest both in the areas of quantum control and quantum information, depends on the specific interaction, and allows one to compare in an operationally meaningful manner any two nonlocal gates. In the case of a two-qubit system, an analytical expression for the interaction cost of any unitary operation given any coupling Hamiltonian is obtained. One gate may be more time consuming than another for any possible interaction. This defines a partial order structure in the set of nonlocal gates, that compares their degree of nonlocality. We analytically characterize this partial order in a region of the set of two-qubit gates.