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Infinite quantum signal processing for arbitrary Szegő functions

Michel Alexis, Lin Lin, Gevorg Mnatsakanyan, Christoph Thiele, Jiasu Wang

TL;DR

This work solves the infinite quantum signal processing problem for Szegő functions by marrying nonlinear Fourier analysis with a Riemann-Hilbert–Weiss (RH–Weiss) algorithm that computes individual phase factors Ψ_k independently. It proves existence and uniqueness of the phase sequence for every f in the Szegő class 𝔖, together with a nonlinear Plancherel identity and a Lipschitz stability bound, enabling robust reconstruction of the target function from phase data. The RH–Weiss framework computes each phase factor via solving a linear system tied to a RH factorization, yielding a polynomial-time principal cost of O(d⁴) for d phase factors and a bit complexity of Ō(log(d/(η ε))). Numerical experiments demonstrate stability and accuracy on high-precision targets, including small η, and compare favorably with Newton-type methods at large discretizations. Overall, the approach provides a provably stable, scalable route to implement iQSP for a broad class of Szegő functions with potential broad impact on quantum signal processing, NLFT-based representations, and spectral-theoretic algorithm design.

Abstract

We provide a complete solution to the problem of infinite quantum signal processing for the class of Szegő 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 Szegő function. The proof of stability involves solving a Riemann-Hilbert factorization problem in nonlinear Fourier analysis using elements of spectral theory.

Infinite quantum signal processing for arbitrary Szegő functions

TL;DR

This work solves the infinite quantum signal processing problem for Szegő functions by marrying nonlinear Fourier analysis with a Riemann-Hilbert–Weiss (RH–Weiss) algorithm that computes individual phase factors Ψ_k independently. It proves existence and uniqueness of the phase sequence for every f in the Szegő class 𝔖, together with a nonlinear Plancherel identity and a Lipschitz stability bound, enabling robust reconstruction of the target function from phase data. The RH–Weiss framework computes each phase factor via solving a linear system tied to a RH factorization, yielding a polynomial-time principal cost of O(d⁴) for d phase factors and a bit complexity of Ō(log(d/(η ε))). Numerical experiments demonstrate stability and accuracy on high-precision targets, including small η, and compare favorably with Newton-type methods at large discretizations. Overall, the approach provides a provably stable, scalable route to implement iQSP for a broad class of Szegő functions with potential broad impact on quantum signal processing, NLFT-based representations, and spectral-theoretic algorithm design.

Abstract

We provide a complete solution to the problem of infinite quantum signal processing for the class of Szegő 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 Szegő function. The proof of stability involves solving a Riemann-Hilbert factorization problem in nonlinear Fourier analysis using elements of spectral theory.
Paper Structure (27 sections, 27 theorems, 319 equations, 3 figures, 2 algorithms)

This paper contains 27 sections, 27 theorems, 319 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Main results
  4. Related works
  5. Discussion and open questions
  6. Preliminaries
  7. Hardy functions
  8. Nonlinear Fourier analysis
  9. Weiss algorithm for constructing a given b
  10. Existing results on Riemann-Hilbert factorization
  11. Riemann-Hilbert-Weiss algorithm for finding phase factors
  12. Outline
  13. Weiss algorithm
  14. Riemann-Hilbert factorization
  15. Numerical experiment
  16. ...and 12 more sections

Key Result

Theorem 1

For each $f \in \mathbf{S}$, there exists a unique sequence $\Psi \in \mathbf{P}$ such that both the $L^2$ convergence criterion in eqn:L2_conv and the nonlinear Plancherel identity in plancherel hold. Furthermore, given $0 < \eta < \frac{1}{2}$, for two functions $f, f' \in \mathbf{S}_{\eta}$ with

Figures (3)

  • Figure 1: The performance of \ref{['alg:RHW_alg']} and Newton's method to find phase factors for the randomly generated phase factors $\Psi$.
  • Figure 2: The performance of \ref{['alg:RHW_alg']} and Newton's method to find phase factors for the target function $0.999\cos(\tau x)$.
  • Figure 3: The difference $\left\lvert\operatorname{Im}[u_d(x,\Psi)] - f(x)\right\rvert$ over interval $[0,1]$ for the phase factors obtained by \ref{['alg:RHW_alg']} and Newton's method for the target function $0.999\cos(\tau x)$.

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2
  • Lemma 3: AlexisMnatsakanyanThiele2023
  • Theorem 4: Extension of AlexisMnatsakanyanThiele2023
  • proof
  • Theorem 5: Extension of AlexisMnatsakanyanThiele2023, Riemann-Hilbert factorization
  • Lemma 6
  • Lemma 7
  • proof
  • Theorem 8
  • ...and 33 more