Coherent SDP matrix-expectation oracle

Input rational parameters $0<\tau \le 2$ and $0<\delta <1$, a choice of repeated controlled-$Q$ or supplied controlled-power QA13 access with its declared costs, and an explicit finite purification circuit $V_{\rho }$ preparing a unit vector whose reduced state on register $S$ is $\rho$, together with its inverse and any controlled versions. Also supply a finite unitary dilation $W_M$ of the two-outcome POVM $E_{\pm }=(I\pm M)\mathrm{/}2$ for a Hermitian contraction $M$, with an output bit and all work registers. Prepare the purification, apply $W_M$ to $S$, and keep the outcome bit coherent; the resulting finite sampler has values $+1$ and $-1$. Supply this signed sampler’s inverse, controlled calls, good and initial-state reflections, observable access, and all value/output encodings. Apply Amplitude-estimation expectation estimator to this sampler on $[-1,1]$ with rational target $\epsilon =\tau \mathrm{/}2$ and failure probability $\delta$, using the selected access mode. Return its classical estimate $\tilde{\mu }$ together with the constructed coherent signed-sampler circuit and its declared access/cost description. The sampling tolerance $\tau \mathrm{/}2$ reserves the other half of the requested tolerance $\tau$ for a purification-bias allowance in the consuming theorem. No inverse preparation is inferred from a measured postselected Gibbs state.

This procedure is distinct from the flagged-trace ratio route inside the sparse SDP solver; that route is specified by the solver's frozen round factory and trace-label construction. Every purification, observable, inverse, controlled call, dilation, measurement and compilation cost is separately charged.