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Arbeitsgemeinschaft: Quantum Signal Processing and Nonlinear Fourier Analysis

Published Jan 1, 2024• 0 citations
Publisher

Abstract

A pivotal algorithm in quantum signal processing represents a real-valued function as a specific entry in a product of $SU(2)$ matrices, where an argument matrix alternates with a sequence of coefficient matrices. Recently, this framework has been identified as a nonlinear Fourier series, a concept studied across various mathematical disciplines. This Arbeitsgemeinschaft explored the intersection of quantum computing and harmonic analysis, emphasizing the potential of this connection.

DOI
10.34657/32042

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•Aug 31, 2026•0 citations
A pivotal algorithm in quantum signal processing represents a real-valued function as a specific entry in a product of SU(2)SU(2)SU(2) matrices, where an argument matrix alternates with a sequence of coefficient matrices. Recently, this framework has been identified as a nonlinear Fourier series, a concept studied across various mathematical disciplines. This Arbeitsgemeinschaft explored the intersection of quantum computing and harmonic analysis, emphasizing the potential of this connection.
We define a nonlinear Fourier transform which maps sequences of contractive n×nn \times nn×n matrices to SU(2n)SU(2n)SU(2n)-valued functions on the circle T\mathbb{T}T. We characterize the image of finitely supported sequences and square-summable sequences on the half-line, and construct an inverse for SU(2n)SU(2n)SU(2n)-valued functions whose diagonal n×nn \times nn×n blocks are outer matrix functions. As an application, we relate this nonlinear Fourier transform with quantum signal processing over U(2n)U(2n)U(2n) and multivariate quantum signal processing.
Quantum signal processing (QSP) provides a representation of scalar polynomials of degree ddd as products of matrices in SU(2)\mathrm{SU}(2)SU(2), parameterized by (d+1)(d+1)(d+1) real numbers known as phase factors. QSP is the mathematical foundation of quantum singular value transformation (QSVT), which is often regarded as one of the most important quantum algorithms of the past decade, with a wide range of applications in scientific computing, from Hamiltonian simulation to solving linear systems of equations and eigenvalue problems. In this article we survey recent advances in the mathematical and numerical analysis of QSP. In particular, we focus on its generalization beyond polynomials, the computational complexity of algorithms for phase factor evaluation, and the numerical stability of such algorithms. The resolution to some of these problems relies on an unexpected interplay between QSP, nonlinear Fourier analysis on SU(2)\mathrm{SU}(2)SU(2), fast polynomial multiplications, and Gaussian elimination for matrices with displacement structure.
The nonlinear Fourier transform (NLFT) extends the classical Fourier transform by replacing addition with matrix multiplication. While the NLFT on SU(1,1)\mathrm{SU}(1,1)SU(1,1) has been widely studied, its SU(2)\mathrm{SU}(2)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 SU(2)\mathrm{SU}(2)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).
This research investigates the application of quantum computing to the verification of matrix operations, focusing on the design of a novel quantum verification framework. The core claim is that leveraging quantum computation offers superior verification accuracy compared to classical methods. The research will explore the establishment of quantum algorithms specifically tailored to assess key quantum properties of matrices. This work aims to enhance the fidelity of quantum computations through rigorous verification and ultimately contribute to the development of more robust and reliable quantum algorithms. The investigation will culminate in a detailed analysis of the framework's potential impact across various matrix-related applications.