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Mathematical and numerical analysis of quantum signal processing

Lin Lin

TL;DR

This survey synthesizes the mathematical and numerical analysis of quantum signal processing (QSP), clarifying when phase factors exist and are unique, and unveiling deep connections to nonlinear Fourier analysis on SU(2). It covers infinite QSP (iQSP) and convergence (L1 and L2) via Szegő theory, complementary polynomials via the Weiss algorithm, and robust inverse NLFT algorithms (layer stripping, Riemann–Hilbert factorization, inverse nonlinear FFT) with stability analyses. It also surveys iterative phase-factor algorithms, the pivotal role of QSP in quantum singular value transformation (QSVT), and practical applications to Hamiltonian simulation, linear systems, and eigenvalue problems, as well as multiple generalizations and open directions. The work highlights how displacement-structure insights, outer function choices, and NLFT theory jointly enable efficient, stable representations and computational pipelines for polynomial and non-polynomial function mappings on quantum hardware.

Abstract

Quantum signal processing (QSP) provides a representation of scalar polynomials of degree $d$ as products of matrices in $\mathrm{SU}(2)$, parameterized by $(d+1)$ 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 $\mathrm{SU}(2)$, fast polynomial multiplications, and Gaussian elimination for matrices with displacement structure.

Mathematical and numerical analysis of quantum signal processing

TL;DR

This survey synthesizes the mathematical and numerical analysis of quantum signal processing (QSP), clarifying when phase factors exist and are unique, and unveiling deep connections to nonlinear Fourier analysis on SU(2). It covers infinite QSP (iQSP) and convergence (L1 and L2) via Szegő theory, complementary polynomials via the Weiss algorithm, and robust inverse NLFT algorithms (layer stripping, Riemann–Hilbert factorization, inverse nonlinear FFT) with stability analyses. It also surveys iterative phase-factor algorithms, the pivotal role of QSP in quantum singular value transformation (QSVT), and practical applications to Hamiltonian simulation, linear systems, and eigenvalue problems, as well as multiple generalizations and open directions. The work highlights how displacement-structure insights, outer function choices, and NLFT theory jointly enable efficient, stable representations and computational pipelines for polynomial and non-polynomial function mappings on quantum hardware.

Abstract

Quantum signal processing (QSP) provides a representation of scalar polynomials of degree as products of matrices in , parameterized by 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 , fast polynomial multiplications, and Gaussian elimination for matrices with displacement structure.

Paper Structure

This paper contains 25 sections, 11 theorems, 72 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. Examples of QSP representations.
  3. Chebyshev polynomials:
  4. All-zero function:
  5. Trigonometric functions:
  6. Inverse function:
  7. Existence and uniqueness of phase factors.
  8. Connections to nonlinear Fourier analysis in SU(2).
  9. Infinite quantum signal processing and convergence properties.
  10. L1 convergence.
  11. L2 convergence.
  12. Complementary polynomials and Weiss algorithm.
  13. Algorithms for inverse nonlinear Fourier transform.
  14. Layer stripping algorithm.
  15. Riemann--Hilbert factorization algorithm.
  16. ...and 10 more sections

Key Result

Theorem 3.1

For any $P, Q \in \mathbb{C}[x]$, positive integer $d$ such that there exists a set of phase factors $\Psi := (\psi_0, \cdots, \psi_d) \in [-\pi, \pi)^{d+1}$ such that

Figures (2)

  • Figure 2.1: QSP representation of $f(x)=\frac{1}{2} \cos(100 x)$ using an even polynomial $p(x)$ of degree $150$. Left: the target function and the QSP representation of $p(x)$. Middle: Error between $p(x)$ and its QSP representation. Right: phase factors after removing a factor of $\pi/4$ on both ends plotted on a log scale.
  • Figure 2.2: Approximating $f(x)=\frac{1}{2\kappa x}$ on $[\kappa^{-1},1]$ using an odd polynomial $p(x)$ of degree $101$ and its QSP representation. Left: the target function and the QSP representation of $p(x)$. Middle: Error between $p(x)$ and its QSP representation. Right: phase factors after removing a factor of $\pi/4$ on both ends plotted on a log scale.

Theorems & Definitions (11)

  • Theorem 3.1: Quantum signal processing GilyenSuLowEtAl2019
  • Corollary 3.2: Quantum signal processing with real target polynomials GilyenSuLowEtAl2019
  • Theorem 3.3: Quantum signal processing with symmetric phase factors WangDongLin2022
  • Theorem 4.1: NLFT bijection alexis2024quantum, tsai2005nlft
  • Lemma 4.2: Connection between QSP and NLFT
  • Theorem 5.1: Infinite QSP DongLinNiEtAl2024_iqsp
  • Theorem 5.2: DongLinNiEtAl2024_iqsp
  • Theorem 5.3: Infinite QSP, $L^2$ convergence alexis2024quantum
  • Theorem 5.4: Infinite QSP, $L^2$ convergence for all Szegő functions alexis2024infinite
  • Lemma 6.1: Nonlinear Plancherel inequality alexis2024infinite
  • ...and 1 more