Asymptotically Optimal Quantum Amplitude Estimation by Generalized Qubitization

Abstract: We first show that the standard deviation error of quantum amplitude estimation is asymptotically lower bounded by approximately $1.28 L^{-1}$, where $L$ is the number of queries. Then we propose a generalized qubitization that can block-encode several polynomial functions simultaneously, and show how it can help estimating quantum amplitude to achieve the optimal asymptotic accuracy, so the bound is tight.