  We propose a computationally efficient method to derive the unitary evolution
that a quantum state is most sensitive to. This allows one to determine the
optimal use of an entangled state for quantum sensing, even in complex systems
where intuition from canonical squeezing examples breaks down. In this paper we
show that the maximal obtainable sensitivity using a given quantum state is
determined by the largest eigenvalue of the quantum Fisher information matrix
(QFIM) and, importantly, the corresponding evolution is uniquely determined by
the coinciding eigenvector. Since we optimize the process of parameter encoding
rather than focusing on state preparation protocols, our scheme is relevant for
any quantum sensor. This procedure naturally optimizes multiparameter
estimation by determining, through the eigenvectors of the QFIM, the maximal
set of commuting observables with optimal sensitivity.
