Two exact quantum signal processing results
Bjorn K. Berntson, Christoph Sünderhauf · 2024
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.