Quantum systems theory and applications to quantum simulation under collective controls - an invitation

We present a unified framework of Lie-algebraic quantum systems theory. It provides a powerful yet straight-forward access to settle problems of controllability, observability, and the design of universal quantum hardware. A particular focus is on the symmetry principles of quantum simulation. They govern under which conditions and to which extent spin systems, fermionic systems, and bosonic systems can mutually simulate oneanother. Finally, we give an explorative outline of quantum dynamics under collective Hamiltonian controls meant to invite further research.