Complementary polynomials in quantum signal processing

TL;DR

This paper introduces a novel complex analysis approach to construct complementary polynomials in quantum signal processing, enabling explicit error control and efficient computation via FFT, outperforming existing numerical methods.

Contribution

The paper develops a contour integral representation for complementary polynomials and an FFT-based algorithm with explicit error guarantees, advancing quantum signal processing techniques.

Findings

The new method provides explicit error bounds for polynomial construction.

The FFT-based algorithm is more efficient than previous optimization methods.

Numerical results show improved performance over existing approaches.

Abstract

Quantum signal processing is a framework for implementing polynomial functions on quantum computers. To implement a given polynomial $P$, one must first construct a corresponding complementary polynomial $Q$. Existing approaches to this problem employ numerical methods that are not amenable to explicit error analysis. We present a new approach to complementary polynomials using complex analysis. Our main mathematical result is a contour integral representation for a canonical complementary polynomial. On the unit circle, this representation has a particularly simple and efficacious Fourier analytic interpretation, which we use to develop a Fast Fourier Transform-based algorithm for the efficient calculation of $Q$ in the monomial basis with explicit error guarantees. Numerical evidence that our algorithm outperforms the state-of-the-art optimization-based method for computing complementary polynomials is provided.