  In this work, we present a lower bound on the quantum Fisher information
(QFI) which is efficiently computable on near-term quantum devices. This bound
itself is of interest, as we show that it satisfies the canonical criteria of a
QFI measure. Specifically, it is essentially a QFI measure for subnormalized
states, and hence it generalizes the standard QFI in this sense. Our bound
employs the generalized fidelity applied to a truncated state, which is
constructed via the $m$ largest eigenvalues and their corresponding
eigenvectors of the probe quantum state $\rho_{\theta}$. Focusing on unitary
families of exact states, we analyze the properties of our proposed lower
bound, and demonstrate its utility for efficiently estimating the QFI.
