Coherent query-to-circuit cost translation

For a QA29 execution, group its complete value-rotated sampler preparations into the distinct supplied unitaries $A_j$, and let $N_j$ be the number of fresh standalone preparations of $A_j$ and $Q_j$ the number of controlled Grover rounds using it. Also let $V_t$ range over the supplied unrotated sampler preparations used for coarse measurements, with $N_t^0$ such fresh preparations. If each controlled Grover round for $A_j$ is the explicit product of one controlled $A_j$ call, one controlled $A_j^{†}$ call, one controlled initial-state reflection, one controlled good-state reflection, and its control phase, then the exact primitive-call cost is ${\sum }_j{N}_j{C}_{A_j,\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{p}}+{\sum }_t{N}_t^0{C}_{V_t,\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{p}}+{\sum }_j{Q}_j(C_{A_j,c}+C_{A_j^{†},c}+C_{S_{0,j},c}+C_{S_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d},j},c}+C_{\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e},j}),$ where every $C$ is the declared cost of that operation. The preparation counts exclude the controlled $A_j$ and $A_j^{†}$ occurrences inside Grover rounds, which are already included in the $Q_j$ terms. With a single $A_q$, no unrotated coarse preparations, and $N_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{p}}=N_q$, this reduces to the homogeneous formula $N_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{p}}{C}_{A_q,\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{p}}+Q(C_{A_q,c}+C_{A_q^{†},c}+C_{S_0,c}+C_{S_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}},c}+C_{\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}})$. Value decoding, band comparisons, reversible arithmetic, controlled rotations, uncomputation, phase estimation, ancilla workspace, circuit description, and classical output/median computations add their own declared gate, width, bit, or oracle costs. If a complete reflection is instead supplied as a named oracle, charge that oracle's declared cost separately and do not also expand it into its components. No bare query bound becomes a gate bound without these implementations.