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 $\mid P(z)\mid \le 1$ on the complex unit circle $\mathbb{T}$ and (ii) a complementary polynomial $Q(z)$, which satisfies $\mid P(z){\mid }^2+\mid Q(z){\mid }^2=1$ on $\mathbb{T}$. We present two exact mathematical results within this context. First, we obtain an exact expression for a certain uniform polynomial approximant of $1\mathrm{/}x$, 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.