Real bounded polynomial extraction in QSVT

By [F1], choose a QSP-admissible pair $P,Q$ with $\mathrm{Re}P=p$. By [F2], choose a phase list $\mathrm{\Phi }$ whose ordered signal product has upper-left entry $P_{\mathrm{\Phi }}=P$. The real part identity is therefore exact for every real $x$.

Let control value $0$ select $\mathrm{\Phi }$ and value $1$ select $-\mathrm{\Phi }$. In each slot with phase $\varphi$ about $\mathrm{\Pi }$, apply the toggle $X_{\mathrm{\Pi }}$ from [F5], then unconditional $R_z(2\varphi )$, then a control-one-controlled $R_z(-4\varphi )$, and then $X_{\mathrm{\Pi }}$ again. In branch $0$ the middle rotation is $R_z(2\varphi )$; in branch $1$ it is $R_z(-2\varphi )$. The toggle places the clean ancilla in $\mathrm{\mid }1⟩$ on $\mathrm{ran}\mathrm{\Pi }$ and in $\mathrm{\mid }0⟩$ on $\ker \mathrm{\Pi }$, so the two system phases are $e^{i\varphi (2\mathrm{\Pi }-I)}$ and $e^{-i\varphi (2\mathrm{\Pi }-I)}$, respectively, with the ancilla returned to $\mathrm{\mid }0⟩$. Thus each slot realizes the required $\mathrm{\Phi }\mathrm{/}-\mathrm{\Phi }$ phase pair. By [F6] and [F10], the signal calls between slots are shared unconditioned calls, and the full controlled unitary has branch operators $U_{\mathrm{\Phi }}$ and $U_{-\mathrm{\Phi }}$. Projecting $\mathrm{\mid }+⟩$ to $\mathrm{\mid }+⟩$ gives their arithmetic mean, since $⟨+\mathrm{\mid }b⟩⟨b\mathrm{\mid }+⟩=1\mathrm{/}2$ for each $b\in \{0,1\}$.

Since each entry of $W_R(x)$ is real and $\overline{Z(\varphi )}=Z(-\varphi )$, entrywise conjugation of the ordered product in [F3] gives $U_{-\mathrm{\Phi }}(x)=\overline{U_{\mathrm{\Phi }}(x)}$. Thus its upper-left polynomial is $P_{-\mathrm{\Phi }}(x)=\overline{P_{\mathrm{\Phi }}(x)}$ for every $x\in [-1,1]$.

Fix a positive singular pair $({\sigma }_j,v_j,u_j)$. By [F4], the two branch compressions contribute coefficients $P_{\mathrm{\Phi }}({\sigma }_j)$ and $P_{-\mathrm{\Phi }}({\sigma }_j)$. Their mean is $\mathrm{Re}{P}_{\mathrm{\Phi }}({\sigma }_j)=p({\sigma }_j)$ by steps 1.1–2.1. For even $d$, [F4] gives the same mean $p(0)$ on each vector in $\ker A$; for odd $d$, $p(0)=0$ and the output block annihilates that kernel. Thus the averaged compression has exactly the stated singular-value transform on all of $H_R$.

There are $d$ alternating signal positions and $d+1$ phase slots by [F7]. The signal list is unconditioned and shared. At each slot, the two toggles use two controlled-reflection calls, and the middle uses one unconditional $R_z(2\varphi )$ plus one branch-controlled $R_z(-4\varphi )$. By [F9], the controlled rotations and reflection calls are supplied and costed separately. For $d=0$, there is one paired phase slot and no signal call, so the same count applies.