SDP matrix-expectation estimation error and cost

Let $M=M^{\ast }$ with $\mathrm{\parallel }M\mathrm{\parallel }\le 1$, and let the coherent signed sampler from Coherent SDP matrix-expectation oracle be supplied. For additive tolerance $0<\tau \le 2$ and failure $0<\delta <1$, supply rational $\tau \mathrm{/}2<t\le \tau$ and $\delta \mathrm{/}2<\beta \le \delta$, and run that procedure at inputs $(t,\beta )$. The QA29 estimator returns an estimate of $\mathrm{tr}(M\rho )$ within $\tau$ with probability at least $1-\delta$ in either declared QA13 access mode. In repeated controlled-$Q$ access it uses $O({\tau }^{-1}(1+\ln (1\mathrm{/}\delta )))$ coherent sampler/reflection rounds; in supplied controlled-power access sum the separately declared power-query costs. When the supplied purification is itself an approximate Gibbs purification at vector error $\xi \le \tau \mathrm{/}4$, the expectation bias is at most $2\xi$; use its rational sampling tolerance $t\mathrm{/}2\le \tau \mathrm{/}2$ to keep total error at most $\tau$. In an SDP iteration, use a union bound across all $m$ observables and charge each Gibbs preparation/inverse, observable access, controlled call, and state/phase implementation.