  With optimal control theory, we compute the maximum possible quantum Fisher
information about the interaction parameter for a Kitaev chain with tunable
long-range interactions in the many-particle Hilbert space. We consider a wide
class of decay laws for the long-range interaction and develop rigorous
asymptotic analysis for the scaling of the quantum Fisher information with
respect to the number of lattice sites. In quantum metrology nonlinear
many-body interactions can enhance the precision of quantum parameter
estimation to surpass the Heisenberg scaling, which is quadratic in the number
of lattice sites. Here for the estimation of the long-range interaction
strength, we observe the Heisenberg to super-Heisenberg transition in such a
$linear$ model, related to the slow decaying long-range correlations in the
model. Finally, we show that quantum control is able to improve the prefactor
rather than the scaling exponent of the quantum Fisher information. This is in
contrast with the case where quantum control has been shown to improve the
scaling of quantum Fisher information with the probe time. Our results clarify
the role of quantum controls and long-range interactions in many-body quantum
metrology.
