Quantum counting by uniform-state amplitude estimation

Input. A Boolean predicate $f:\{0,1{\}}^n\to \{0,1\}$, an integer $n\ge 0$, $N=2^n$, an explicitly supplied controlled membership reflection for $S_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$, a power of two $M=2^m\ge 2$, and the other controlled accesses in Amplitude estimation and counting input/output contract.

Prepare $N^{-1\mathrm{/}2}{\sum }_x\mathrm{\mid }x⟩=H^{\otimes n}\mathrm{\mid }{0}^n⟩$, run the $m$-qubit QPE circuit for $Q$ with $M=2^m$ labels, measure $y$, and output the exact algebraic fraction estimate $\tilde{a}={\sin }^2(\pi y\mathrm{/}M)$ and count estimate $\tilde{t}=N\tilde{a}$. For $M\ge 4$ this is the QFT amplitude-estimation circuit of QFT-based amplitude estimation; for $M=2$, use its one-phase-qubit QPE circuit directly, without invoking the $M\ge 4$ error theorem. The fraction and count are real-valued algebraic outputs; integer rounding is optional postprocessing. The controlled membership reflection is an explicit access requirement. In repeated-unitary mode each controlled $Q$ call charges one controlled membership reflection; in power-query mode each supplied controlled power carries its declared cost vector, including any membership-reflection implementations it uses. Neither access mode is inferred from the bare phase oracle $\mathrm{\mid }x⟩\mapsto (-1{)}^{f(x)}$.