  Quantum optimal control has numerous important applications ranging from
pulse shaping in magnetic-resonance imagining to laser control of chemical
reactions and quantum computing. Our objective is to address two major
challenges that have limited the success of applications of quantum optimal
control so far: non-commutativity inherent in quantum systems and non-convexity
of quantum optimal control problems involving more than three quantum levels.
Methodologically, we address the non-commutativity of the control Hamiltonian
at different times by the use of Magnus expansion. To tackle the non-convexity,
we employ non-commutative polynomial optimisation and non-commutative geometry.
As a result, we present the first globally convergent methods for quantum
optimal control.
