Matrix-function transformation error and resource bound

On a Hermitian input $A$, each active parity singular-value transform in [F4] equals ordinary polynomial functional calculus. Indeed, for an eigenvector with eigenvalue $x>0$, choose its left and right singular vectors equal, so its transform coefficient is $p_j(x)$. For $x<0$, choose singular value $\mathrm{\mid }x\mathrm{\mid }$ and left vector the negative of the right vector; the coefficient is $-p_j(\mathrm{\mid }x\mathrm{\mid })=p_j(x)$ when $p_j$ is odd, while for even parity it is $p_j(\mathrm{\mid }x\mathrm{\mid })=p_j(x)$. At $x=0$, an odd polynomial vanishes and the even transform assigns $p_j(0)$ on the kernel. The spectral resolution [F7] and basis-independence in [F4] make these equalities independent of choices within degenerate eigenspaces. Also [F1] gives $\mathrm{\mid }f(\lambda )\mathrm{\mid }\le {\sum }_{j\in J}{\gamma }_j=B$ on $\sigma (H)$; apply [F2] to the zero polynomial and $f$ to obtain $\mathrm{\parallel }f(H)\mathrm{\parallel }\le B$.

By [F2], $\mathrm{\parallel }{p}_j(A)-g_j(H)\mathrm{/}{\gamma }_j\mathrm{\parallel }\le {\tau }_j$, since $A=H\mathrm{/}\alpha$ and the scalar bound holds on its spectrum. By [F3]–[F4], replacing the exact signal block $A$ by $\tilde{A}$ changes the ideal transform $p_j^{\mathrm{S}\mathrm{V}}$ by at most $4d_j\sqrt{{\epsilon }_{BE}}$; step 1.1 identifies the transform at $A$ with $p_j(A)$. The implemented component adds at most ${\kappa }_j$ by its definition, including numerical phase and signal implementation errors. For any unit vector $v$, the vector triangle inequality in [F9] bounds the action of the total difference by the sum of these three operator bounds; taking the supremum over unit $v$ gives $\mathrm{\parallel }{Y}_j-g_j(H)\mathrm{/}{\gamma }_j\mathrm{\parallel }\le {\tau }_j+4d_j\sqrt{{\epsilon }_{BE}}+{\kappa }_j$, where $Y_j$ is its actual top-left block.

The exact LCU block for these $Y_j$ is $B^{-1}{\sum }_{j\in J}{c}_j{\gamma }_j{Y}_j$ by [F5]; its actual circuit differs from that block by at most ${\kappa }_{LCU}$. The decomposition in [F1] gives $f(H)={\sum }_{j\in J}{c}_j{g}_j(H)$ on the spectrum. For each unit vector $v$, apply the vector triangle inequality [F9] to the weighted sum of component errors from step 2.1 and the LCU implementation error; taking the supremum over $v$ gives the displayed operator-norm bound after multiplying each component bound by ${\gamma }_j\mathrm{/}B$.

The exact signal-call count for component $j\in J$ is $d_j$ by [F8]; a SELECT branch invokes its designated component sequence, so its selected-component branch has at most ${\max }_{j\in J}{d}_j$ signal calls. The implemented SELECT circuit may contain the sum of separately controlled component occurrences; its actual oracle-occurrence count remains separately charged. The remaining resource coordinates are the declared preparation, inverse, control, coefficient, phase, reflection, compilation, and output costs in [F1], [F6], and [F8]; neither a degree count nor an exact phase list supplies those costs automatically.