Quantum counting as an expectation special case

Input. A Boolean domain $X=\{0,1{\}}^n$ with $N=2^n$, a predicate $f:X\to \{0,1\}$, a phase-register size $M=2^m$ for an integer $m\ge 2$ (so $M\ge 4$), and the uniform coherent access and controlled membership reflection required by Quantum counting by uniform-state amplitude estimation. Let $S=\{x:f(x)=1\}$.

Use the unchanged QA13 counting estimator Quantum counting by uniform-state amplitude estimation with this $M$ and its supplied access model, and return its exact algebraic fraction estimate $\tilde{\mu }=\tilde{t}\mathrm{/}N={\sin }^2(\pi y\mathrm{/}M)$ for the expectation task. Its real-valued count estimate $\tilde{t}=N\tilde{\mu }$ may also be reported, with count and expectation outputs explicitly distinguished. Do not round to an integer unless that additional postprocessing and its cost are explicitly requested. By Uniform counting is finite indicator expectation, this is the QA29 expectation task for the uniform law and observable $g={\mathbf{1}}_S$; the applicable count guarantee is Quantum counting error and boundary cases, whose phase-size hypothesis $M\ge 4$ is met. Divide its count-error radius by $N$ for the expectation-error radius, with the same success event and probability. This item defines no second amplitude-estimation or counting routine.