Inverse nonlinear fast Fourier transform on SU(2) with applications to quantum signal processing

TL;DR

This paper introduces a fast, stable algorithm for the inverse nonlinear Fourier transform on SU(2), enabling efficient quantum signal processing applications with near-linear complexity.

Contribution

It develops the first numerically stable, fast inverse nonlinear Fourier transform algorithm on SU(2), applicable to quantum signal processing.

Findings

Established numerical stability of the layer stripping algorithm.

Developed a near-linear complexity inverse nonlinear FFT algorithm.

Applied the algorithm to quantum signal processing tasks.

Abstract

The nonlinear Fourier transform (NLFT) extends the classical Fourier transform by replacing addition with matrix multiplication. While the NLFT on $\mathrm{SU}(1,1)$ has been widely studied, its $\mathrm{SU}(2)$ variant has only recently attracted attention due to emerging applications in quantum signal processing (QSP) and quantum singular value transformation (QSVT). In this paper, we investigate the inverse NLFT on $\mathrm{SU}(2)$ and establish the numerical stability of the layer stripping algorithm for the first time under suitable conditions. Furthermore, we develop a fast and numerically stable algorithm, called inverse nonlinear fast Fourier transform, for performing inverse NLFT with near-linear complexity. This algorithm is applicable to computing phase factors for both QSP and the generalized QSP (GQSP).