Amplitude-estimation expectation estimator

Input. A finite expectation-estimation task with an exact encoded observable $g:X\to [a,b]$, a coherent sampler $V$ for its law, rational target $\epsilon >0$, and rational failure probability $0<\delta <1$. For $a<b$, require $\epsilon \le b-a$. The supplied access model must include every forward, inverse, controlled, value, predicate, and reflection operation that the chosen QA13 amplitude-estimation circuit invokes.

If $a=b$, return $a$. Otherwise compute the exact affine value $q(x)=\frac{g(x)-a}{b-a}\in [0,1]\mathrm{.}$ Append a good flag to the coherent sampler. Reversibly compute $q(x)$ in the declared value workspace, apply the supplied one-qubit rotation $\mathrm{\mid }0⟩⟼\sqrt{1-q(x)}\mathrm{\mid }0⟩+\sqrt{q(x)}\mathrm{\mid }1⟩$ controlled by the sample/value registers, and uncompute all scratch used only for $q$. Let $A_q$ denote the resulting state-preparation unitary, including $V$ and the controlled rotation; let $P_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$ project onto flag $\mathrm{\mid }1⟩$. Run the unchanged median-confidence estimator Median confidence amplification for amplitude estimation on $(A_q,P_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}})$ with amplitude tolerance $\eta =\epsilon \mathrm{/}(b-a)$ and failure probability $\delta$, and call its estimate $\tilde{q}$. Return $\tilde{\mu }=a+(b-a)\tilde{q}\mathrm{.}$

For $a=b$, the exact-return branch has no quantum call. When $a<b$, the input is a costed finite circuit/oracle description. Charge each use of $A_q$, $A_q^{†}$, controlled versions, the good and initial-state reflections, value decoding, reversible arithmetic and uncomputation, the controlled rotation and its precision, phase estimation, state preparation, gates, width, ancillas, and classical output processing according to Circuit-family and computation cost model. No operation is free because it is described by the law $p$ or observable $g$.