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

Hongkang Ni, Rahul Sarkar, Lexing Ying, Lin Lin

TL;DR

This work provides a numerically stable framework for the inverse SU(2) NLFT, bridging nonlinear Fourier analysis with quantum signal processing. It introduces a provably stable layer stripping method under the condition that $a^*(z)$ has no zeros in the closed unit disk and a near-linear time inverse NLFT, $ ext{O}(n \, ext{log}^2 n)$, via a fast inverse NLFFT inspired by displacement-structured matrices. The results unify QSP and generalized QSP (GQSP) within NLFT, enabling efficient, stable computation of phase factors for both frameworks, and establish local Lipschitz continuity of the inverse NLFT map. These contributions advance stable, scalable phase-factor computation critical for quantum algorithms and QSVT implementations, with clear criteria on when the stability guarantees hold (outer $a^*$) and how precision scales with problem size. The work also clarifies the fundamental role of displacement structure in stability analyses and provides explicit error bounds and bit-precision requirements for practical numeric accuracy.

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).

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

TL;DR

This work provides a numerically stable framework for the inverse SU(2) NLFT, bridging nonlinear Fourier analysis with quantum signal processing. It introduces a provably stable layer stripping method under the condition that has no zeros in the closed unit disk and a near-linear time inverse NLFT, , via a fast inverse NLFFT inspired by displacement-structured matrices. The results unify QSP and generalized QSP (GQSP) within NLFT, enabling efficient, stable computation of phase factors for both frameworks, and establish local Lipschitz continuity of the inverse NLFT map. These contributions advance stable, scalable phase-factor computation critical for quantum algorithms and QSVT implementations, with clear criteria on when the stability guarantees hold (outer ) and how precision scales with problem size. The work also clarifies the fundamental role of displacement structure in stability analyses and provides explicit error bounds and bit-precision requirements for practical numeric accuracy.

Abstract

The nonlinear Fourier transform (NLFT) extends the classical Fourier transform by replacing addition with matrix multiplication. While the NLFT on has been widely studied, its 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 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).
Paper Structure (28 sections, 25 theorems, 188 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 28 sections, 25 theorems, 188 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related works
  4. Outline
  5. Notation
  6. Nonlinear Fourier transform and its structure
  7. NLFT
  8. Complementary polynomials
  9. Inverse NLFT
  10. Connections between NLFT, QSP, and GQSP
  11. Quantum signal processing
  12. Generalized quantum signal processing
  13. Correspondence between QSP, GQSP, and NLFT
  14. Layer stripping algorithm and inverse nonlinear fast Fourier transform
  15. Layer stripping algorithm for inverse NLFT
  16. ...and 13 more sections

Key Result

Lemma 2.1

Let $\ell(m,n)$ be the space of all compactly supported sequences $\boldsymbol{\gamma}: \mathbb{Z} \rightarrow \mathbb{C}$ supported on the interval $m \leq k \leq n$, in the strict sense that $\gamma_m, \gamma_n \neq 0$. Let $\boldsymbol{\gamma} \in \ell(m,n)$ and $\overbrace{\boldsymbol{\gamma}} =

Figures (1)

  • Figure 1: (A) The residual $\|\overbrace{\boldsymbol{\gamma}} - \overbrace{\hat{\boldsymbol{\gamma}}}\|_{L^{\infty}(\mathbb{T})}$ for different degree $n$. (B) For $n=80$, the error for each component $\gamma_k$.

Theorems & Definitions (44)

  • Lemma 2.1: NLFT structure
  • Lemma 2.2: NLFT bijection
  • Lemma 2.3: No zeros in $\overline{\mathbb{D}}$ property
  • Lemma 3.1
  • proof
  • Theorem 3.2: QSP-NLFT correspondence
  • proof
  • Theorem 3.3: GQSP-NLFT correspondence
  • proof
  • Lemma 4.1: Layer stripping properties
  • ...and 34 more