  We investigate the universality of multi-spin systems in architectures of
various symmetries of coupling type and topology. Explicit reachability sets
under symmetry constraints are provided. Thus for a given (possibly symmetric)
experimental coupling architecture several decision problems can be solved in a
unified way: (i) can a target Hamiltonian be simulated? (ii) can a target gate
be synthesised? (iii) to which extent is the system observable by a given set
of detection operators? and, as a special case of the latter, (iv) can an
underlying system Hamiltonian be identified with a given set of detection
operators? Finally, in turn, lack of symmetry provides a convenient necessary
condition for full controllability. Though often easier to assess than the
well-established Lie-algebra rank condition, this is not sufficient unless the
candidate dynamic simple Lie algebra can be pre-identified uniquely, which is
fortunately less complicated than expected.

Quantum Physics