  We present the experimental measurement, on a quantum processor, of a series
of polynomial lower bounds that converge to the quantum Fisher information
(QFI), a fundamental quantity for certifying multipartite entanglement that is
useful for metrological applications. We combine advanced methods of the
randomized measurement toolbox to obtain estimators that are robust against
drifting errors caused uniquely during the randomized measurement protocol. We
estimate the QFI for Greenberg-Horne-Zeilinger states, observing genuine
multipartite entanglement. Then, we prepare the ground state of the transverse
field Ising model at the critical point using a variational circuit. We
estimate its QFI and investigate the interplay between state optimization and
noise induced by increasing the circuit depth.
