  We consider a quantum system with a time-independent Hamiltonian parametrized
by a set of unknown parameters $\alpha$. The system is prepared in a general
quantum state by an evolution operator that depends on a set of unknown
parameters $P$. After the preparation, the system evolves in time, and it is
characterized by a time-dependent observable ${\cal O}(t)$. We show that it is
possible to obtain closed-form expressions for the gradients of the distance
between ${\cal O}(t)$ and a calculated observable with respect to $\alpha$, $P$
and all elements of the system density matrix, whether for pure or mixed
states. These gradients can be used in projected gradient descent to infer
$\alpha$, $P$ and the relevant density matrix from dynamical observables. We
combine this approach with random phase wave function approximation to obtain
closed-form expressions for gradients that can be used to infer population
distributions from averaged time-dependent observables in problems with a large
number of quantum states participating in dynamics. The approach is illustrated
by determining the temperature of molecular gas (initially, in thermal
equilibrium at room temperature) from the laser-induced time-dependent
molecular alignment.
