  Optimal quantum control of continuous variable systems poses a formidable
computational challenge because of the high-dimensional character of the system
dynamics. The framework of quantum invariants can significantly reduce the
complexity of such problems, but it requires the knowledge of an invariant
compatible with the Hamiltonian of the system in question. We explore the
potential of a Gaussian invariant that is suitable for quadratic Hamiltonians
with any given number of motional degrees of freedom for quantum optimal
control problems that are inspired by current challenges in
ground-state-to-ground-state shuttling of trapped-ions.
