Two exact quantum signal processing results

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 $|P(z)|\le 1$ on the complex unit circle $\mathbb{T}$ and (ii) a complementary polynomial $Q(z)$, which satisfies $|P(z){|}^2+|Q(z){|}^2=1$ 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 $1/x$, 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.

This conference paper announces two results. The first of these, on matrix inversion polynomials, will be elaborated elsewhere. The second result, on complementary polynomials, is presented fully in arXiv:2406.04246 [quant-ph] and will appear in Communications in Mathematical Physics