  The Quantum Fisher Information (QFI) plays a crucial role in quantum
information theory and in many practical applications such as quantum
metrology. However, computing the QFI is generally a computationally demanding
task. In this work we introduce a lower bound on the QFI called the sub-Quantum
Fisher Information (sub-QFI), which can be efficiently estimated on a quantum
computer for an $n$-qubit state using $2n$ qubits. The sub-QFI is based on the
super-fidelity, an upper bound on Uhlmann's fidelity. We analyze the sub-QFI in
the context of unitary families, where we derive several crucial properties
including its geometrical interpretation. In particular, we prove that the QFI
and the sub-QFI are maximized for the same optimal state, which implies that
the sub-QFI is faithful to the QFI in the sense that both quantities share the
same global extrema. Based on this faithfulness, the sub-QFI acts as an
efficiently computable surrogate for the QFI for quantum sensing and quantum
metrology applications. Finally, we provide additional meaning to the sub-QFI
as a measure of coherence, asymmetry, and purity loss.
