  Sensing a classical signal using a linear quantum device is a pervasive
application of quantum-enhanced measurement. The fundamental precision limits
of linear waveform estimation, however, are not fully understood. In certain
cases, there is an unexplained gap between the known waveform-estimation
Quantum Cram\'er-Rao Bound and the optimal sensitivity from quadrature
measurement of the outgoing mode from the device. We resolve this gap by
establishing the fundamental precision limit, the waveform-estimation Holevo
Cram\'er-Rao Bound, and how to achieve it using a nonstationary measurement. We
apply our results to detuned gravitational-wave interferometry to accelerate
the search for post-merger remnants from binary neutron-star mergers. If we
have an unequal weighting between estimating the signal's power and phase, then
we propose how to further improve the signal-to-noise ratio by a factor of
$\sqrt2$ using this nonstationary measurement.
