  There is a prevalent effort to achieve quantum-enhanced metrology using
criticality. However, the extent to which estimation precision is enhanced
through criticality still needs further exploration under the constraint of
finite time resources. We clarify relations between quantum metrology and
criticality through a unitary parametrization process with a Hamiltonian
governed by su(1, 1) Lie algebra. We demonstrate that the determination of the
generator in the parameterization can be treated as an extended brachistochrone
problem. Furthermore, the dynamic quantum Fisher information about the
parameter exhibits a power-law dependence on the evolution time as the system
approaches its critical point. By investigating the dynamic sensing proposals
of three quantum critical systems, we show that the asymptotic behavior of
sensitivity is consistent with our predictions. Our theory provides a deep
understanding on the interplay of quantum metrology and criticality, providing
insights into the underlying connections that involve both quantum phenomena
and classical problems.
