  Arrays of atoms trapped in optical tweezers combine features of programmable
analog quantum simulators with atomic quantum sensors. Here we propose
variational quantum algorithms, tailored for tweezer arrays as programmable
quantum sensors, capable of generating entangled states on-demand for precision
metrology. The scheme is designed to generate metrological enhancement by
optimizing it in a feedback loop on the quantum device itself, thus preparing
the best entangled states given the available quantum resources. We apply our
ideas to generate spin-squeezed states on Sr atom tweezer arrays, where
finite-range interactions are generated through Rydberg dressing. The
complexity of experimental variational optimization of our quantum circuits is
expected to scale favorably with system size. We numerically show our approach
to be robust to noise, and surpassing known protocols.
