Variational quantum amplitude estimation

We propose to perform amplitude estima-tion with the help of constant-depth quantum circuits that variationally approximate states during amplitude amplification. In the con -text of Monte Carlo (MC) integration, we nu-merically show that shallow circuits can ac-curately approximate many amplitude ampli-fication steps. We combine the variational ap-proach with maximum likelihood amplitude es-timation [Y. Suzuki et al., Quantum Inf. Pro-cess. 19, 75 (2020)] in variational quantum amplitude estimation (VQAE). VQAE typi-cally has larger computational requirements than classical MC sampling. To reduce the variational cost, we propose adaptive VQAE and numerically show in 6 to 12 qubit sim-ulations that it can outperform classical MC sampling.