Quantum SDP query bound and QRAM cost boundary

Write $Q=\max (1,Rr\mathrm{/}\epsilon )$. On the nontrivial branch of [F1], $T=\tilde{O}(Q^2)$, $\eta =\tilde{O}(Q^{-1})$ and ${\theta }^{-1}=O(Q)$. Its rational finite data and certified implementations meet [F3]'s hypotheses. That supplier's aggregate bound is $\tilde{O}(\sqrt{nm}{s}^2(Rr\mathrm{/}\epsilon {)}^8)$, already including derived sparse-row construction, spectral shifts, all coherent estimator/minimum caps and the outer threshold calls. Since $Rr\mathrm{/}\epsilon \le Q$, it is at most the claimed general query bound.

For diagonal inputs, [F3] exports its separate scalar flag construction's aggregate query bound $\tilde{O}(\sqrt{nm}(Rr\mathrm{/}\epsilon {)}^5)$. The inequality $Rr\mathrm{/}\epsilon \le Q$ gives $\tilde{O}(\sqrt{nm}{Q}^5)$. That supplier includes finite scalar exponential and probability-rotation arithmetic, frozen approximate minima, the common-normalization ratio, and all threshold repetitions. This profile uses that separate diagonal construction rather than substituting $s=1$ in the general bound.

The output guarantee is [F2]. No further factor for the outer search, Fourier frequency count or sparse-row construction multiplies the aggregate query bound in step 1.1: those costs are already included in [F3]. Nontrivial accuracy has $Rr\mathrm{/}\epsilon >1\mathrm{/}2$, so replacing it by $Q$ only changes a fixed constant there. The boundary branches of [F1] precede the positive-$R$ formulas and retain their stated direct output costs.

Let $F_t$ be once-only factory generation/certification/storage, $N_{flag,t}=\tilde{O}(\sqrt{nm}\mathrm{/}\theta )$ and $N_{shift,t}=\tilde{O}(\sqrt{n})$ the complete preparation/inverse call counts, and $G_{flag,t},G_{shift,t}$ their actual nonquery gate prices excluding reference preparation. Let $A_t$ charge the remaining coherent AE/QFT arithmetic, rounding, division, comparison, histories, capped-minimum coins and classical control; let $G_{QRAM}$ and $W$ charge actual optional memory and output/record operations. [F3] gives the actual additive record ${\sum }_t\text{\hspace{0.17em}⁣}[F_t+N_{flag,t}(C_{\mathrm{\Phi }}+G_{flag,t})+N_{shift,t}(C_{\mathrm{\Phi }}+G_{shift,t})+A_t]+G_{QRAM}+W,$ with the threshold repetition factor. Include input-query gate prices when translating queries to gates, and avoid double counting when a primitive already includes reference preparation. The summed reference term is $\tilde{O}(\sqrt{nm}{Q}^3{C}_{\mathrm{\Phi }})$; the Gibbs coefficient factory has the following direct count. At round $t$, [F1] bounds the exponent norm by $\eta t$, so its shifted Gibbs interval has radius $K_t=O(1+\eta t)$ and extension margin $1\mathrm{/}2$. The cutoffs in [F5] have Taylor length $\tilde{O}(K_t)$ and arcsine degree $\tilde{O}(K_t^2)$. Computing each truncated convolution by its finite double sum, then all signed-binomial accumulations, takes $\tilde{O}(K_t^5)$ arithmetic operations. Summing gives $\tilde{O}(T+{\eta }^5{T}^6)=\tilde{O}(Q^7)$; actual bit and phase-generation costs remain separately charged. Coherent operations in $A_t$ are quantum gates/work even when a classical description of their arithmetic is available.

In the power-of-two case [F4] gives the explicit $O(\log n)$ reference circuit. In general retain the actual repeated $C_{\mathrm{\Phi }}$ price and any separately allocated preparation error. The unit-cost QRAM profile and a direct reversible sorted-row implementation have different actual prices, each covered by step 2.2. Therefore the source's same-order gate profile follows precisely if these prices fit its stated envelope; otherwise step 1.1 and step 1.2 are query bounds with the explicit additional record of step 2.2. This retains the rich source scope and its computational-model qualification without treating expensive exact arithmetic or an undeclared oracle as free.