Bernoulli mean as a good-state probability

Here $\mu =(1-p)\cdot 0+p\cdot 1=p$ by [F1] and the two-point law [F3]. The range is $[a,b]=[0,1]$, so [F4] gives $q(x)=x$.

Conditional on sample $x$, the controlled rotation places good probability $x$. Averaging over the sample law, the total good probability is $(1-p)\cdot 0+p\cdot 1=p=\mu$.

Thus the QA13 amplitude estimator is estimating the Bernoulli mean itself; [F5] yields additive error $\epsilon$ with confidence $1-\delta$ in either declared access mode. In repeated controlled-$Q$ access it uses $O({\epsilon }^{-1}(1+\ln (1\mathrm{/}\delta )))$ controlled sampler/reflection rounds; in supplied controlled-power access its power queries retain their separately declared prices. The circuit costs remain those declared by the supplied sampler.