  The quantum Fisher information (QFI) represents a fundamental concept in
quantum physics. On the one hand, it quantifies the metrological potential of
quantum states in quantum-parameter-estimation measurements. On the other hand,
it is intrinsically related to the quantum geometry and multipartite
entanglement of many-body systems. Here, we explore how the QFI can be
estimated via randomized measurements, an approach which has the advantage of
being applicable to both pure and mixed quantum states. In the latter case, our
method gives access to the sub-quantum Fisher information, which sets a lower
bound on the QFI. We experimentally validate this approach using two platforms:
a nitrogen-vacancy center spin in diamond and a 4-qubit state provided by a
superconducting quantum computer. We further perform a numerical study on a
many-body spin system to illustrate the advantage of our randomized-measurement
approach in estimating multipartite entanglement, as compared to quantum state
tomography. Our results highlight the general applicability of our method to
general quantum platforms, including solid-state spin systems, superconducting
quantum computers and trapped ions, hence providing a versatile tool to explore
the essential role of the QFI in quantum physics.
