  With an ever-expanding ecosystem of noisy and intermediate-scale quantum
devices, exploring their possible applications is a rapidly growing field of
quantum information science. In this work, we demonstrate that variational
quantum algorithms feasible on such devices address a challenge central to the
field of quantum metrology: The identification of near-optimal probes and
measurement operators for noisy multi-parameter estimation problems. We first
introduce a general framework which allows for sequential updates of
variational parameters to improve probe states and measurements and is widely
applicable to both discrete and continuous-variable settings. We then
demonstrate the practical functioning of the approach through numerical
simulations, showcasing how tailored probes and measurements improve over
standard methods in the noisy regime. Along the way, we prove the validity of a
general parameter-shift rule for noisy evolutions, expected to be of general
interest in variational quantum algorithms. In our approach, we advocate the
mindset of quantum-aided design, exploiting quantum technology to learn close
to optimal, experimentally feasible quantum metrology protocols.
