  Quantum sensors based on single Nitrogen-Vacancy (NV) defects in diamond are
state-of-the-art tools for nano-scale magnetometry with precision scaling
inversely with total measurement time $\sigma_{B} \propto 1/T$ (Heisenberg
scaling) rather than as the inverse of the square root of $T$, with $\sigma_{B}
=1/\sqrt{T}$ the Shot-Noise limit. This scaling can be achieved by means of
phase estimation algorithms (PEAs) using adaptive or non-adaptive feedback, in
combination with single-shot readout techniques. Despite their accuracy, the
range of applicability of PEAs is limited to periodic signals involving single
frequencies with negligible temporal fluctuations. In this Letter, we propose
an alternative method for precision magnetometry in frequency multiplexed
signals via compressive sensing (CS) techniques. We show that CS can provide
for precision scaling approximately as $\sigma_{B} \approx 1/T$, both in the
case of single frequency and frequency multiplexed signals, as well as for a
5-fold increase in sensitivity over dynamic-range gain, in addition to reducing
the total number of resources required.
