  In this study, we address a control-constrained optimal control problem
pertaining to the transformation of quantum states. Our objective is to
navigate a quantum system from an initial state to a desired target state while
adhering to the principles of the Liouville-von Neumann equation. To achieve
this, we introduce a cost functional that balances the dual goals of fidelity
maximization and energy consumption minimization. We derive optimality
conditions in the form of the Pontryagin Maximum Principle (PMP) for the
matrix-valued dynamics associated with this problem. Subsequently, we present a
time-discretized computational scheme designed to solve the optimal control
problem. This computational scheme is rooted in an indirect method grounded in
the PMP, showcasing its versatility and efficacy. To illustrate the
practicality and applicability of our methodology, we employ it to address the
case of a spin $\frac{1}{2}$ particle subjected to interaction with a magnetic
field. Our findings shed light on the potential of this approach to tackle
complex quantum control scenarios and contribute to the broader field of
quantum state transformations.
