Abstract
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 P(z)≤ 1 on the complex unit circle T and (ii) a complementary polynomial Q(z), which satisfies P(z)²+ Q(z)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.