Amplitude Table Precision and Compilation

Each of the $2D$ real coordinates differs by at most $2^{-b}$, so $\mathrm{\parallel }v-a{\mathrm{\parallel }}_2^2\le 2D{2}^{-2b}$ and $\mathrm{\parallel }v-a{\mathrm{\parallel }}_2\le \eta$. The reverse triangle inequality gives $\mathrm{\mid }\mathrm{\parallel }v{\mathrm{\parallel }}_2-1\mathrm{\mid }\le \eta$, hence $\mathrm{\parallel }v{\mathrm{\parallel }}_2\ge 1-\eta >0$.

Use [F8] to choose a rational ${\epsilon }_c^{\mathrm{\prime }}$ with ${\epsilon }_c\mathrm{/}2<{\epsilon }_c^{\mathrm{\prime }}<{\epsilon }_c$. Let the exact normalized-table circuit be $U_G\cdots U_1$. Each block has positive arity $r_j\le k$ and, by [A1], an exact circuit with at most $C_k$ CNOTs. Scan that circuit chronologically, maintaining one pending one-qubit product on each wire. At a one-qubit gate, compose it into its wire's pending product. Before each CNOT, emit the two pending products on its control and target and reset them to identity; pending products on other wires commute with that CNOT and may remain pending. At the end emit all $r_j$ pending products. This preserves the exact circuit product and uses at most $2C_k+r_j\le M_k$ one-qubit factors, counting identities if necessary, so the one-qubit count is positive. Apply [A2] to this factorization with real target error ${\epsilon }_c^{\mathrm{\prime }}\mathrm{/}G$; it gives a compiled block ${\tilde{U}}_j$ and a unit scalar ${\lambda }_j$ with $\mathrm{\parallel }{U}_j-{\lambda }_j{\tilde{U}}_j{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {\epsilon }_c^{\mathrm{\prime }}\mathrm{/}G<{\epsilon }_c\mathrm{/}G$. Each block includes all of its controls and targets before compilation; thus ${\lambda }_j$ is a scalar on its entire support and becomes one global scalar when tensored with identities on the other wires.

Split the normalization error through $v$: by [F5], $\mathrm{\parallel }v\mathrm{/}\mathrm{\parallel }v{\mathrm{\parallel }}_2-a{\mathrm{\parallel }}_2\le \mathrm{\parallel }v\mathrm{/}\mathrm{\parallel }v{\mathrm{\parallel }}_2-v{\mathrm{\parallel }}_2+\mathrm{\parallel }v-a{\mathrm{\parallel }}_2=\mathrm{\mid }1-\mathrm{\parallel }v{\mathrm{\parallel }}_2\mathrm{\mid }+\mathrm{\parallel }v-a{\mathrm{\parallel }}_2\le 2\eta$.

Insert the $G$ intermediate products between $U_G\cdots U_1$ and $({\prod }_j{\lambda }_j){\tilde{U}}_G\cdots {\tilde{U}}_1$ to obtain the telescoping sum with one block difference in each term. Each support-level block difference, when extended by identity to the full ordered register, has operator norm at most its support norm: after permuting the product basis to group support and unused wires, write a vector as ${\sum }_t{x}_t\otimes e_t$ in an orthonormal basis $(e_t)$ of the unused wires, and use ${\sum }_t\mathrm{\parallel }A{x}_t{\mathrm{\parallel }}^2\le \mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^2{\sum }_t\mathrm{\parallel }{x}_t{\mathrm{\parallel }}^2$. The wire permutation preserves norm by [F9]. For any unit input vector, [F5] bounds the norm of the telescoping sum by the sum of term norms; [F6] gives norm $1$ for every surrounding unitary factor, so [F7] bounds each term by ${\epsilon }_c^{\mathrm{\prime }}\mathrm{/}G$. Taking the maximum over unit inputs with [F6] gives whole-circuit operator error at most ${\sum }_{j=1}^G{\epsilon }_c^{\mathrm{\prime }}\mathrm{/}G={\epsilon }_c^{\mathrm{\prime }}<{\epsilon }_c$. Applying this to the fixed initial vector gives projective chord error at most ${\epsilon }_c^{\mathrm{\prime }}$ from the exact normalized-$v$ output.

By [F3], choose a phase aligning the compiled output to the exact normalized-$v$ joint output within ${\epsilon }_c^{\mathrm{\prime }}$, and a phase aligning that exact output to $a\otimes \mathrm{\mid }{0}^s⟩$ within $2\eta$ by step 2.1. Multiplying the second phase by the first and applying [F5] gives $d_{\mathrm{c}\mathrm{h}}([\mathrm{o}\mathrm{u}\mathrm{t}],[a\otimes \mathrm{\mid }{0}^s⟩])\le 2\eta +{\epsilon }_c^{\mathrm{\prime }}<2\eta +{\epsilon }_c$.

For fixed maximum arity $k$, [A2] uses $O(M_k{\log }^c(M_{k}G\mathrm{/}{\epsilon }_c^{\mathrm{\prime }})+C_k)$ finite-alphabet gates per block at accuracy ${\epsilon }_c^{\mathrm{\prime }}\mathrm{/}G$. Since $M_k,C_k$ are constants, ${\epsilon }_c^{\mathrm{\prime }}>{\epsilon }_c\mathrm{/}2$, and $G\mathrm{/}{\epsilon }_c>2$, this is $O({\log }^c(G\mathrm{/}{\epsilon }_c))$ per block; hence $G$ blocks use $O(G{\log }^c(G\mathrm{/}{\epsilon }_c))$ gates. If the rational input table $v$ has an encoding as in [F4], it contains $2Db$ bits, and [F1] reads every real/imaginary coordinate in its preprocessing pass before exact normalization; this input scan is separate from quantum gate count. No angle-computation or compiler bit-time follows from these bounds.