Amplitude estimation

Given a state-preparation oracle $U$ such that $U\ket{0}=\ket{\psi}=\sqrt{p}\ket{\psi_1}+\sqrt{1-p}\ket{\psi_0}$ with $\langle\psi_1|\psi_0\rangle=0$, and a reflection oracle $R_1=I-2\ket{\psi_1}\!\bra{\psi_1}$. When $p(1-p)=\Omega(1)$, there exists a quantum algorithm that estimates $p$ up to precision $\epsilon$ with failure probability at most $\eta$, using $\mathcal{O}(\log(1/\epsilon)+\log(1/\eta))$ ancilla qubits, $\mathcal{O}(1/(\epsilon\eta))$ queries to $U$, $U^\dagger$, and controlled-$R_1$, and $\mathcal{O}\!\left(n/(\epsilon\eta)+(\log(1/\epsilon)+\log(1/\eta))^2\right)$ elementary gates.