First-order query and gate cost

For the term-list input $H={\sum }_{j=1}^m{H}_j$ in Term-list Hamiltonian access, let $t$ be a finite time code in its expression language, let $0<ϵ<1$ be rational, and let $C_2\ge 0$ be a certified rational satisfying $C_2\ge {\sum }_{1\le j<\mathrm{\ell }\le m}\mathrm{\parallel }[H_j,H_{\mathrm{\ell }}]\mathrm{\parallel }$. Supply a certified rational upper bound $R\ge t^2{C}_2\mathrm{/}ϵ$ and put $r=\max \{1,\lceil R\rceil \}$ and $h=t\mathrm{/}r$. When $t\mathrm{\ne }0$, suppose each term call at the exact signed code $h$ is supported and declared, directly or through separately declared inverse or time-reversed access, with at most $q_j$ named queries and $g_j$ gate occurrences. Let $Q_{\mathrm{w}\mathrm{r}\mathrm{a}\mathrm{p}}$ and $G_{\mathrm{w}\mathrm{r}\mathrm{a}\mathrm{p}}$ be the separately declared query and gate costs of uniform circuit composition and generation. For $t\mathrm{\ne }0$, the algorithm First-order term-list simulator uses at most $r{\sum }_j{q}_j+Q_{\mathrm{w}\mathrm{r}\mathrm{a}\mathrm{p}}$ queries and $r{\sum }_j{g}_j+G_{\mathrm{w}\mathrm{r}\mathrm{a}\mathrm{p}}$ gates. Its exact term product has error at most $ϵ\mathrm{/}2$. If, when $t\mathrm{\ne }0$, implemented calls are unitary and their total operator-norm error over all $rm$ calls is at most $ϵ\mathrm{/}2$, the returned unitary is within $ϵ$ of $U_H(t)$. When $t=0$, the algorithm returns the identity with no term calls. Coefficient, description-bit, ancilla, and classical-generation resources remain separate coordinates.