Segmented Taylor error and resources

On the identity branch let $S=\mathrm{\Lambda }\mathrm{\mid }t\mathrm{\mid }\le ϵ\mathrm{/}4\le 1\mathrm{/}4<1$. In an orthonormal eigenbasis as in [F2], each Hamiltonian eigenvalue satisfies $\mathrm{\mid }\lambda \mathrm{\mid }\le \mathrm{\Lambda }$. Its scalar-series error obeys $\mathrm{\mid }{e}^{-it\lambda }-1\mathrm{\mid }\le {\sum }_{k\ge 1}{S}^k\mathrm{/}k!\le {\sum }_{k\ge 1}{S}^k=S\mathrm{/}(1-S)\le ϵ\mathrm{/}(4-ϵ)\le ϵ\mathrm{/}3$. The diagonal norm calculation in [F2] proves $\mathrm{\parallel }I-U_H(t)\mathrm{\parallel }\le ϵ\mathrm{/}3$, and the error is zero if $t=0$ or $\mathrm{\Lambda }=0$. Otherwise step 2 chooses a finite $r$ with $x:=\mathrm{\Lambda }\mathrm{\mid }t\mathrm{\mid }\mathrm{/}r\le \ln 2$; no lower bound on $S$ is needed. Hence $s_K\le e^x\le 2$, and cancelling padding makes the norm exactly $2$ without changing $M_K$.

By [F3], the padded circuit has block $M_K(h)\mathrm{/}2$. The uncorrected amplified block is $-3M_K(h)\mathrm{/}2+M_K(h)M_K(h{)}^{†}{M}_K(h)\mathrm{/}2$ by [F4], so the explicit phase correction in [L1] gives the corresponding opposite-sign block; [F5] supplies the needed full isometry estimate including leakage. The Taylor-tail rule gives $\mathrm{\parallel }{M}_K(h)-U_H(h)\mathrm{\parallel }\le \eta$, and $\eta \le 1\mathrm{/}1600<1\mathrm{/}100$. Apply [F5] with $V=U_H(h)$ to obtain a clean-input isometry error at most $10\sqrt{\eta }=ϵ\mathrm{/}(4r)$ for each segment, including any finite reference and all earlier retained registers.

By [L1], the scheduler uses the positive rational ${\eta }_{-}:={ϵ}_{-}^2\mathrm{/}(1600r^2)$, where $\eta \mathrm{/}4\le {\eta }_{-}\le \eta$. The loop terminates because $n!$ tends to infinity. Since ${ϵ}_{-}\le ϵ\le 1$ and $r\ge 1$, ${\eta }_{-}\le 1\mathrm{/}1600$; hence $L_{-}:=\ln (4\mathrm{/}{\eta }_{-})>1$. For $N=\lceil 16e{L}_{-}\mathrm{/}\ln (e+L_{-})\rceil$, the inequalities $\ln (e+L_{-})\le \sqrt{e+L_{-}}$ and $e+L_{-}<4L_{-}$ give $N\mathrm{/}e>4\sqrt{e+L_{-}}$, hence $\ln (N\mathrm{/}e)>\frac{1}{2}\ln (e+L_{-})$. Therefore $N\ln (N\mathrm{/}e)\ge 8e{L}_{-}\ge L_{-}+\ln 4$. Monotonicity of $\ln s$ gives $\ln (N!)={\sum }_{k=1}^N\ln k\ge {\int }_1^N\ln sds=N\ln N-N+1$, so $N!\ge (N\mathrm{/}e{)}^N\ge 4\mathrm{/}{\eta }_{-}$. The least power of two at least $N$ is less than $2N$ and already meets the factorial comparison, so the scheduler stops at some $n<2N$. Thus $K=n-1=O(1+L_{-}\mathrm{/}\ln (e+L_{-}))$. Since $\eta \mathrm{/}4\le {\eta }_{-}\le \eta$ and $\eta \le 1\mathrm{/}1600$, this is $O(1+\log (1\mathrm{/}\eta )\mathrm{/}\log \log (e+1\mathrm{/}\eta ))$. The rule uses only a finite integer loop, and its cost is a separate classical coordinate. For the error estimates, $e^x{x}^n\mathrm{/}n!\le 2\mathrm{/}n!\le {\eta }_{-}\mathrm{/}2\le \eta \mathrm{/}2$ because $x\le \ln 2<1$.

The ideal segment maps each input to the same system evolution $U_H(h)$ and a fresh zero register. Telescoping the $r$ compositions, every intervening exact segment is an isometry and has norm $1$, so the total access-circuit isometry error is at most $rϵ\mathrm{/}(4r)=ϵ\mathrm{/}4$. The ideal system factor is $U_H(h{)}^r=U_H(t)$ by [F6].

Replacing the exact access calls by their compiled unitaries changes the full circuit in operator norm by at most the sum of their per-call errors: expand the product difference as a telescoping sum and use norm $1$ for each unitary factor. The declared sum is at most $ϵ\mathrm{/}2$ by [L1]. The triangle inequality therefore gives $\mathrm{\parallel }{V}_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}}{J}_{\mathrm{i}\mathrm{n}}-J_{\mathrm{o}\mathrm{u}\mathrm{t}}{U}_H(t)\mathrm{\parallel }\le ϵ\mathrm{/}4+ϵ\mathrm{/}2=3ϵ\mathrm{/}4\le ϵ$.

First take a pure unit input, including any finite reference. The segmented isometry bound in step 3.1, or the identity-branch bound in step 1.1 tensored with that reference, gives vector error at most $ϵ$. For the identity branch, write a system-reference vector as ${\sum }_j{x}_j\otimes f_j$ in an orthonormal reference basis [F15]; [F11] makes its squared error the sum ${\sum }_j\mathrm{\parallel }(I-U_H(t))x_j{\mathrm{\parallel }}^2\le (ϵ\mathrm{/}3{)}^2{\sum }_j\mathrm{\parallel }{x}_j{\mathrm{\parallel }}^2$, justifying that extension explicitly. Then [F12] bounds the joint pure-state trace distance by $ϵ$. The difference of the two joint density operators is Hermitian, so applying partial-trace contractivity [F13] over all work registers preserves this bound.

For a general input density operator, [F8]–[F9] give an orthonormal eigenbasis $(e_j)$ with nonnegative eigenvalues ${\lambda }_j$; [F14] and trace one imply ${\sum }_j{\lambda }_j=1$. Take an auxiliary reference space with orthonormal basis $(f_j)$, which exists by [F15], and set $\mathrm{\Psi }={\sum }_j\sqrt{{\lambda }_j}{e}_j\otimes f_j$. By the product inner product [F11], $\mathrm{\parallel }\mathrm{\Psi }{\mathrm{\parallel }}^2={\sum }_j{\lambda }_j=1$. In the product basis, its projector has entries $X_{(i,j),(i^{\mathrm{\prime }},j^{\mathrm{\prime }})}=\sqrt{{\lambda }_i{\lambda }_{i^{\mathrm{\prime }}}}{\mathbf{1}}_{i=j}{\mathbf{1}}_{i^{\mathrm{\prime }}=j^{\mathrm{\prime }}}$. The coordinate rule [F10] gives $({\mathrm{Tr}}_{\mathrm{r}\mathrm{e}\mathrm{f}}X{)}_{i{i}^{\mathrm{\prime }}}={\sum }_j{X}_{(i,j),(i^{\mathrm{\prime }},j)}$, which is ${\lambda }_i$ if $i=i^{\mathrm{\prime }}$ and $0$ otherwise, exactly the matrix of $\rho$. Thus $\mathrm{\Psi }$ purifies $\rho$. Apply step 4.1 with the original reference and this purifying reference, then trace out the work registers and the purifying register; [F13] again preserves the bound. Therefore the reduced outputs for $\rho$ differ in trace distance by at most $ϵ$.

In each segment the chronological OAA circuit has two forward $W$ factors, one $W^{†}$ factor, and two reflections [L1]. Expanding each $W=P^{†}{S}_{\mathrm{s}\mathrm{e}\mathrm{l}}P$ gives three PREPARE and three PREPARE${}^{†}$ calls, two SELECT and one SELECT${}^{†}$ call per segment. In the indexed term-query implementation [L1], a length-$k$ tuple acts by processing its labels in reverse order, so its $K$ conditionally active positions implement exactly $V_{j_1}\cdots V_{j_k}$. The phase and invalid-label rules are separately specified. Thus each SELECT or its adjoint uses at most $K$ controlled indexed term calls and the total is at most $3rK$. If only individual controlled $V_j$ are used to assemble a multiplexer, apply the $j$th unitary conditioned on label equality, for every $j<m$: exactly one block is active on each valid label, so this implements the same direct sum with at most $m$ individual calls per indexed call. This gives $3rmK$ individual calls with the declared label/routing/phase costs; no such decomposition bound is inferred for an arbitrary supplied SELECT. The expanded coefficient list has $N_K={\sum }_{k=0}^K{m}^k$ ordered terms and two retained padding labels, giving the stated widths. Internal work and all other resource coordinates are charged as in [F7]. The identity branch has no queries or retained segment registers.