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