Two exact quantum signal processing results

Quantum signal processing (QSP) is a framework for implementing certain polynomial functions via quantum circuits. To construct a QSP circuit, one needs (i) a target polynomial $P$ ($z$), which must satisfy $\vert P (z) \vert$ ≤ 1 on the complex unit circle T and (ii) a complementary polynomial $Q$ ($z$), which satisfies $\vert P(z)\vert^{2}+ \vert Q(z)\vert^{2}= 1$ on T. We present two exact mathematical results within this context. First, we obtain an exact expression for a certain uniform polynomial approximant of $1/ x$, which is used to perform matrix inversion via quantum circuits. Second, given a generic target polynomial $P (z)$, we construct the complementary polynomial $Q(z)$ exactly via integral representations, valid throughout the entire complex plane.