  Quantum sensors can show unprecedented sensitivities, provided they are
controlled in a very specific, optimal way. Here, we consider a spin sensor of
time-varying fields in the presence of dephasing noise, and we show that the
problem of finding the pulsed control field that optimizes the sensitivity
(i.e., the smallest detectable signal) can be mapped to the determination of
the ground state of a spin chain. We find an approximate but analytic solution
of this problem, which provides a \emph{lower bound} for the sensitivity and a
pulsed control very close to optimal, which we further use as initial guess for
realizing a fast simulated annealing algorithm. We experimentally demonstrate
the sensitivity improvement for a spin-qubit magnetometer based on a
nitrogen-vacancy center in diamond.
