  Although measuring the deterministic waveform of a weak classical force is a
well-studied problem, estimating a random waveform, such as the spectral
density of a stochastic signal field, is much less well-understood despite it
being a widespread task at the frontier of experimental physics.
State-of-the-art precision sensors of random forces must account for the
underlying quantum nature of the measurement, but the optimal quantum protocol
for interrogating such linear sensors is not known. We derive the fundamental
precision limit, the extended channel quantum Cram\'er-Rao bound, and the
optimal protocol that attains it. In the experimentally relevant regime where
losses dominate, we prove that non-Gaussian state preparation and measurements
are required for optimality. We discuss how this non-Gaussian protocol could
improve searches for signatures of quantum gravity, stochastic gravitational
waves, and axionic dark matter.
