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Fast Phase Factor Finding for Quantum Signal Processing

Hongkang Ni, Lexing Ying

TL;DR

Two efficient and stable algorithms for recovering phase factors in quantum signal processing (QSP) are presented, based on nonlinear Fourier analysis and fast solvers for structured matrices and Fast Fixed Point Iteration.

Abstract

This paper presents two efficient and stable algorithms for recovering phase factors in quantum signal processing (QSP), a crucial component of many quantum algorithms. The first algorithm, the ``Half Cholesky" method, which is based on nonlinear Fourier analysis and fast solvers for structured matrices, demonstrates robust performance across all regimes. The second algorithm, ``Fast Fixed Point Iteration," provides even greater efficiency in the non-fully-coherent regime. Both theoretical analysis and numerical experiments demonstrate the significant advantages of these new methods over all existing approaches.

Fast Phase Factor Finding for Quantum Signal Processing

TL;DR

Two efficient and stable algorithms for recovering phase factors in quantum signal processing (QSP) are presented, based on nonlinear Fourier analysis and fast solvers for structured matrices and Fast Fixed Point Iteration.

Abstract

This paper presents two efficient and stable algorithms for recovering phase factors in quantum signal processing (QSP), a crucial component of many quantum algorithms. The first algorithm, the ``Half Cholesky" method, which is based on nonlinear Fourier analysis and fast solvers for structured matrices, demonstrates robust performance across all regimes. The second algorithm, ``Fast Fixed Point Iteration," provides even greater efficiency in the non-fully-coherent regime. Both theoretical analysis and numerical experiments demonstrate the significant advantages of these new methods over all existing approaches.
Paper Structure (16 sections, 4 theorems, 77 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 16 sections, 4 theorems, 77 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Problem Setup
  3. Main results
  4. Related works
  5. Organization of the paper
  6. Fast NLFT algorithm for phase factor finding
  7. Review of NLFT and Riemann-Hilbert-Weiss algorithm
  8. Rearrangement of the linear systems
  9. Exploiting the block structure of $A$
  10. Fast LDL factorization using displacement structure
  11. Complexity Analysis
  12. Stability Analysis
  13. Efficient evaluation of QSP matrix and Fast FPI method
  14. Efficient evaluation of QSP matrix
  15. Application to FPI method: FFPI method
  16. ...and 1 more sections

Key Result

Theorem 1

Assume that $f$ is a $(2d)$-degree even polynomial in $\mathbf{S}_{\eta}$. There exists a deterministic algorithm (alg: phase factor finding) to stably compute the phase factor sequence $\Psi$ to precision $\epsilon$ with a computational cost of $\tilde{\mathcal{O}}\left(d^2 + \frac{d\log(d/(\eta\ep

Figures (2)

  • Figure 1: Example for random generated target polynomial $f$ with $\left\lVert f\right\rVert_{\infty}=0.5$. This figure shows the runtime comparison among the five methods: Newton, FPI, FFPI, Riemann-Hilbert-Weiss, and Half Cholesky method.
  • Figure 2: Example for Hamiltonian simulation, with $\left\lVert f\right\rVert_{\infty} = 0.999$. This figure shows the runtime comparison for the fully-coherent regime among the three methods: Newton, Riemann-Hilbert-Weiss, and Half Cholesky method.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Lemma 4
  • proof