QSVT inverse-state output and success cost

Write the finite SVD $A={\sum }_j{\sigma }_j\mathrm{\mid }{u}_j⟩⟨v_j\mathrm{\mid }$. Since $b={\sum }_j{\beta }_j{u}_j$ is supported on ${\sigma }_j\ge \gamma$, applying the odd transform to ${\bar{A}}^{†}$ maps each $u_j$ to $v_j$ with multiplier given by the verified phase-sequence polynomial. By [F9] this ideal polynomial is exactly the generated completed-pair target before numerical phase approximation; its total difference from the intended $p$ transform, including polynomial shrink and phase/circuit errors, is included in $\zeta$. The target multiplier is $\delta \mathrm{/}(2({\sigma }_j\mathrm{/}\alpha ))=\gamma \mathrm{/}(2{\sigma }_j)$, the coefficient of $F=(\delta \mathrm{/}2){\bar{A}}^{+}$; [F1] and [F3] therefore bound the operator error on the supported subspace by $\zeta$.

Put $v=F\mathrm{\mid }b⟩$ and let $w$ be the implemented success vector. Then $\mathrm{\parallel }w-v\mathrm{\parallel }\le \zeta$. By [F2], $\mathrm{\parallel }v\mathrm{\parallel }\ge \delta \mathrm{/}2$, so reverse triangle inequality gives $\mathrm{\parallel }w\mathrm{\parallel }\ge \delta \mathrm{/}2-\zeta >0$ and $p_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}}=\mathrm{\parallel }w{\mathrm{\parallel }}^2\ge (\delta \mathrm{/}2-\zeta {)}^2$. Moreover, $\parallel \frac{w}{\mathrm{\parallel }w\mathrm{\parallel }}-\frac{v}{\mathrm{\parallel }v\mathrm{\parallel }}\parallel \le \frac{\mathrm{\mid }\mathrm{\parallel }v\mathrm{\parallel }-\mathrm{\parallel }w\mathrm{\parallel }\mathrm{\mid }+\mathrm{\parallel }w-v\mathrm{\parallel }}{\mathrm{\parallel }v\mathrm{\parallel }}\le \frac{2\zeta }{\mathrm{\parallel }v\mathrm{\parallel }}\le \frac{4\zeta }{\delta }\mathrm{.}$ The conditional state is defined because $\mathrm{\parallel }w\mathrm{\parallel }>0$.

By [F2], $p_{\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}}=\mathrm{\parallel }v{\mathrm{\parallel }}^2\ge {\delta }^2\mathrm{/}4$. In the implemented case, if $\zeta \le \delta \mathrm{/}4$, step 1.2 gives $p_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}}\ge {\delta }^2\mathrm{/}16$. By [F4], one attempt uses $m$ block-encoding signal calls, and phase/reflection and controlled-call resources remain separate.

Put $\mathrm{\mid }G⟩=w\mathrm{/}\mathrm{\parallel }w\mathrm{\parallel }$ for the one-attempt good vector. By [F6], after any integer number $j$ of amplitude-amplification iterates the unnormalized good component is $s_j\mathrm{\mid }G⟩$ for a scalar $s_j$. On a positive-probability good branch $s_j\mathrm{\ne }0$, and the normalized ket is $(s_j\mathrm{/}\mathrm{\mid }{s}_j\mathrm{\mid })\mathrm{\mid }G⟩$; its phase can differ from that of $\mathrm{\mid }G⟩$. Each such branch has exactly the density operator $\mathrm{\mid }G⟩⟨G\mathrm{\mid }$, including after the stopping history is forgotten. Thus any returned unit representative is $\psi =c\mathrm{\mid }G⟩$ with $\mathrm{\mid }c\mathrm{\mid }=1$, and choosing $\omega =\bar{c}$ gives $\omega \psi =\mathrm{\mid }G⟩$ and the error bound of step 1.2. This identifies the terminal state ray without rephasing any coherent circuit or controlled call. Fresh rounds use the same preparation circuit, and [F7] returns only on that good measurement; [F8] supplies eventual success for $p_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}}>0$. Applying [F5] to the actual success weight $p_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}}$ gives expected $O(p_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}}^{-1\mathrm{/}2})$ attempts, hence expected $O(m\mathrm{/}\sqrt{p_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}}})$ signal calls. If [F1] is established, substitute $m=O({\delta }^{-1}\ln (1\mathrm{/}e))$; if $\zeta \le \delta \mathrm{/}4$, step 2.1 gives $p_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}}^{-1\mathrm{/}2}\le 4\mathrm{/}\delta$. All other resource coordinates remain separate by [F4]–[F5].