Variational quantum algorithms stand as the most promising approaches towards practical applications of near-term quantum computers [1,2]. However, these methodologies usually require a large number of measurements, which represents an important roadblock for future real-world applications. 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 and show that it is competitive with state-of-the-art measurement reduction approaches. Our computational experiments further suggest a measurement scaling exponent below 2. We also highlight the potential of the informational completeness of the measurement outcomes by reusing the ground-state energy estimation data to perform reduced state tomography with high fidelity on the XX model spin chain [3].

[1] Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alan Aspuru-Guzik,“The theory of variational hybrid quantum-classical algorithms” New J. Phys. 18 023023 (2016).

[2] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay M. Gambetta, “Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets,” Nature 549, 242, (2017).

[3] Guillermo García-Pérez, Matteo A. C. Rossi, Boris Sokolov, Elsi-Mari Borrelli, and Sabrina Maniscalco “Pairwise tomography networks for many-body quantum systems" Phys. Rev. Research 2, 023393 (2020).