QFT-based amplitude estimation

Input. An instance from Amplitude estimation and counting input/output contract, an integer $m\ge 2$, $M=2^m$, and either the declared controlled-power access mode or repeated controlled-$Q$ access.

Instructions. Prepare $\psi =A\mathrm{\mid }0⟩$, run the $M$-label QPE circuit on $Q$ using controlled powers $Q^{2^j}$, $j=0,\dots ,m-1$, and measure its phase register to obtain $y\in \{0,\dots ,M-1\}$. Output the exact cyclotomic value $\tilde{a}={\sin }^2(\pi y\mathrm{/}M)=\frac{2-{\zeta }_M^y-{\zeta }_M^{-y}}{4},{\zeta }_M=e^{2\pi i\mathrm{/}M},$ with the real embedding specified by this root. A decimal approximation is postprocessing and is charged separately. The leading minus in $Q$ is retained: on a controlled call it is the control phase $C(-I)$, not a discarded standalone global phase.