  What can one do with a given tunable quantum device? We provide complete
symmetry criteria deciding whether some effective target interaction(s) can be
simulated by a set of given interactions. Symmetries lead to a better
understanding of simulation and permit a reasoning beyond the limitations of
the usual explicit Lie closure. Conserved quantities induced by symmetries pave
the way to a resource theory for simulability. On a general level, one can now
decide equality for any pair of compact Lie algebras just given by their
generators without determining the algebras explicitly. Several physical
examples are illustrated, including entanglement invariants, the relation to
unitary gate membership problems, as well as the central-spin model.
