  The Quantum Fisher information (QFI) quantifies the ultimate precision of
estimating a parameter from a quantum state, and can be regarded as a
reliability measure of a quantum system as a quantum sensor. However,
estimation of the QFI for a mixed state is in general a computationally
demanding task. In this work we present a variational quantum algorithm called
Variational Quantum Fisher Information Estimation (VQFIE) to address this task.
By estimating lower and upper bounds on the QFI, based on bounding the
fidelity, VQFIE outputs a range in which the actual QFI lies. This result can
then be used to variationally prepare the state that maximizes the QFI, for the
application of quantum sensing. In contrast to previous approaches, VQFIE does
not require knowledge of the explicit form of the sensor dynamics. We simulate
the algorithm for a magnetometry setup and demonstrate the tightening of our
bounds as the state purity increases. For this example, we compare our bounds
to literature bounds and show that our bounds are tighter.
