Two exact quantum signal processing results

TL;DR

This paper presents two exact mathematical results in quantum signal processing, including a polynomial approximant for matrix inversion and a method to construct complementary polynomials exactly.

Contribution

It introduces exact formulas for a polynomial approximant of 1/x and a way to construct the complementary polynomial Q(z) for any target polynomial P(z).

Findings

Derived an exact polynomial approximant for 1/x used in matrix inversion.

Constructed the complementary polynomial Q(z) exactly via integral representations.

Results are valid throughout the entire complex plane.

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