  We consider quantum metrology for unitary evolutions generated by
parameter-dependent Hamiltonians. We focus on the unitary evolutions generated
by the Ising Hamiltonian that describes the dynamics of a one-dimensional chain
of spins with nearest-neighbour interactions and in the pres- ence of a global,
transverse, magnetic field. We analytically solve the problem and show that the
precision with which one can estimate the magnetic field (interaction strength)
given one knows the interaction strength (magnetic field) scales at the
Heisenberg limit, and can be achieved by a linear superposition of the vacuum
and N free fermion states. In addition, we show that GHZ-type states exhibit
Heisenberg scaling in precision throughout the entire regime of parameters.
Moreover, we numerically observe that the optimal precision using a product
input state scales at the standard quantum limit.
