Adaptive measurement strategy for noisy quantum amplitude estimation with variational quantum circuits

In quantum computation, amplitude estimation is a fundamental subroutine that is utilized in various quantum algorithms. A general important task of such estimation problems is to characterize the estimation lower bound, which is referred to as quantum Cram\'er-Rao bound (QCRB), and to construct an optimal estimator that achieves QCRB. This paper studies the amplitude estimation in the presence of depolarizing noise with unknown intensity. The main difficulty in this problem is that the optimal measurement depends on both the unknown quantum state and the amplitude we aim to estimate. To deal with these issues, we utilize the variational quantum circuits to approximate the (unknown) optimal measurement basis combined with the two-step adaptive estimation strategy which was proposed in the quantum estimation theory. We numerically show that the proposed method can nearly attain the QCRB.