  We present adaptive measurement techniques tailored for variational quantum
algorithms on near-term small and noisy devices. In particular, we generalise
earlier "learning to measure" strategies in two ways. First, by considering a
class of adaptive positive operator valued measures (POVMs) that can be
simulated with simple projective measurements without ancillary qubits, we
decrease the amount of required qubits and two-qubit gates. Second, by
introducing a method based on Quantum Detector Tomography to mitigate the
effect of noise, we are able to optimise the POVMs as well as to infer
expectation values reliably in the currently available noisy quantum devices.
Our numerical simulations clearly indicate that the presented strategies can
significantly reduce the number of needed shots to achieve chemical accuracy in
variational quantum eigensolvers, thus helping to solve one of the bottlenecks
of near-term quantum computing.
