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.
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