  Variational quantum algorithms stand as the most promising approaches towards
practical applications of near-term quantum computers. However, these
methodologies usually require a large number of measurements, which represents
an important roadblock for future real-world use cases. We introduce a novel
approach to tackle this problem: a variational measurement scheme. We present
an algorithm that optimises informationally complete POVMs on the fly in order
to minimise the statistical fluctuations in the estimation of relevant cost
functions. We use it in combination with the Variational Quantum Eigensolver to
calculate ground-state energies of molecular Hamiltonians in numerical
simulations and show that it is competitive with state-of-the-art measurement
reduction approaches. We also highlight the potential of the informational
completeness of the measurement outcomes by reusing the ground-state energy
estimation data to perform high-fidelity reduced state tomography.
