  We study when a multipartite non--local unitary operation can
deterministically or probabilistically simulate another one when local
operations of a certain kind -in some cases including also classical
communication- are allowed. In the case of probabilistic simulation and
allowing for arbitrary local operations, we provide necessary and sufficient
conditions for the simulation to be possible. Deterministic and probabilistic
interconversion under certain kinds of local operations are used to define
equivalence relations between gates. In the probabilistic, bipartite case this
induces a finite number of classes. In multiqubit systems, however, two unitary
operations typically cannot simulate each other with non-zero probability of
success. We also show which kind of entanglement can be created by a given
non--local unitary operation and generalize our results to arbitrary operators.
