  It is a general fact that the coupling constant of an interacting many-body
Hamiltonian do not correspond to any observable and one has to infer its value
by an indirect measurement. For this purpose, quantum systems at criticality
can be considered as a resource to improve the ultimate quantum limits to
precision of the estimation procedure. In this paper, we consider the
one-dimensional quantum Ising model as a paradigmatic example of many-body
system exhibiting criticality, and derive the optimal quantum estimator of the
coupling constant varying size and temperature. We find the optimal external
field, which maximizes the quantum Fisher information of the coupling constant,
both for few spins and in the thermodynamic limit, and show that at the
critical point a precision improvement of order $L$ is achieved. We also show
that the measurement of the total magnetization provides optimal estimation for
couplings larger than a threshold value, which itself decreases with
temperature.

Quantum Physics