Infinite quantum signal processing for arbitrary Szeg\H{o} functions

TL;DR

This paper introduces a novel, stable numerical algorithm called the Riemann-Hilbert-Weiss algorithm for infinite quantum signal processing of Szeg ext{"o} functions, enabling independent computation of phase factors.

Contribution

The paper presents the first provably stable numerical method for computing phase factors of Szeg ext{"o} functions, solving a key problem in quantum signal processing.

Findings

The Riemann-Hilbert-Weiss algorithm computes phase factors independently.

The algorithm is the first stable numerical method for Szeg ext{"o} functions.

The approach uses nonlinear Fourier analysis and spectral theory.

Abstract

We provide a complete solution to the problem of infinite quantum signal processing for the class of Szeg\H{o} functions, which are functions that satisfy a logarithmic integrability condition and include almost any function that allows for a quantum signal processing representation. We do so by introducing a new algorithm called the Riemann-Hilbert-Weiss algorithm, which can compute any individual phase factor independent of all other phase factors. Our algorithm is also the first provably stable numerical algorithm for computing phase factors of any arbitrary Szeg\H{o} function. The proof of stability involves solving a Riemann-Hilbert factorization problem in nonlinear Fourier analysis using elements of spectral theory.