Symmetry in quantum system theory: Rules for quantum architecture design

We investigate universality in the sense of controllability and observability, of multi-qubit systems in architectures of various symmetries of coupling type and topology. By determining the respective dynamic system Lie algebras, 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, the absence 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. Thus for architectures with various Ising and Heisenberg coupling types we give design rules sufficient to ensure full controllability. In view of follow-up studies, we relate the unification of necessary and sufficient conditions for universality to filtering simple Lie subalgebras of su(N) comprising classical and exceptional types.