  Quantum metrology has been studied for a wide range of systems with
time-independent Hamiltonians. For systems with time-dependent Hamiltonians,
however, due to the complexity of dynamics, little has been known about quantum
metrology. Here we investigate quantum metrology with time-dependent
Hamiltonians to bridge this gap. We obtain the optimal quantum Fisher
information for parameters in time-dependent Hamiltonians, and show proper
Hamiltonian control is necessary to optimize the Fisher information. We derive
the optimal Hamiltonian control, which is generally adaptive, and the
measurement scheme to attain the optimal Fisher information. In a minimal
example of a qubit in a rotating magnetic field, we find a surprising result
that the fundamental limit of $T^{2}$ time scaling of quantum Fisher
information can be broken with time-dependent Hamiltonians, which reaches
$T^{4}$ in estimating the rotation frequency of the field. We conclude by
considering level crossings in the derivatives of the Hamiltonians, and point
out additional control is necessary for that case.
