  Quantum waveform estimation, in which quantum sensors sample entire time
series, promises to revolutionize the sensing of weak and stochastic signals,
such as the biomagnetic impulses emitted by firing neurons. For long duration
signals with rapid transients, regular quantum sampling becomes prohibitively
resource intensive as it demands many measurements with distinct control and
readout. In this Manuscript, we demonstrate how careful choice of quantum
measurements, along with the modern mathematics of compressive sensing,
achieves quantum waveform estimation of sparse signals in a number of
measurements far below the Nyquist requirement. We sense synthesized
neural-like magnetic signals with radiofrequency-dressed ultracold atoms,
retrieving successful waveform estimates with as few measurements as
compressive theoretical bounds guarantee.
