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Quantum Newton's method for solving system of nonlinear algebraic equations

Cheng Xue, Yu-Chun Wu, Guo-Ping Guo

TL;DR

Through numerical simulation, the complexity of QNM is sublinear with [Formula: see text], which provides quantum advantage compared with the optimal classical algorithm.

Abstract

While quantum computing provides an exponential advantage in solving system of linear equations, there is little work to solve system of nonlinear equations with quantum computing. We propose quantum Newton's method (QNM) for solving $N$-dimensional system of nonlinear equations based on Newton's method. In QNM, we solve the system of linear equations in each iteration of Newton's method with quantum linear system solver. We use a specific quantum data structure and $l_{\infty}$ tomography with sample error $ε_s$ to implement the classical-quantum data conversion process between the two iterations of QNM, thereby constructing the whole process of QNM. The complexity of QNM in each iteration is $O(\log^4N/ε_s^2)$. Through numerical simulation, we find that when $ε_s>>1/\sqrt{N}$, QNM is still effective, so the complexity of QNM is sublinear with $N$, which provides quantum advantage compared with the optimal classical algorithm.

Quantum Newton's method for solving system of nonlinear algebraic equations

TL;DR

Through numerical simulation, the complexity of QNM is sublinear with [Formula: see text], which provides quantum advantage compared with the optimal classical algorithm.

Abstract

While quantum computing provides an exponential advantage in solving system of linear equations, there is little work to solve system of nonlinear equations with quantum computing. We propose quantum Newton's method (QNM) for solving -dimensional system of nonlinear equations based on Newton's method. In QNM, we solve the system of linear equations in each iteration of Newton's method with quantum linear system solver. We use a specific quantum data structure and tomography with sample error to implement the classical-quantum data conversion process between the two iterations of QNM, thereby constructing the whole process of QNM. The complexity of QNM in each iteration is . Through numerical simulation, we find that when , QNM is still effective, so the complexity of QNM is sublinear with , which provides quantum advantage compared with the optimal classical algorithm.

Paper Structure

This paper contains 16 sections, 1 theorem, 42 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Newton's Method
  3. QNM
  4. Quantum data structure
  5. Preprocessing
  6. Processing
  7. Postprocessing
  8. Complexity
  9. Query complexity
  10. Time Complexity
  11. Application and Numerical Simulation
  12. Problems and difference format
  13. Nonlinear first-order diffusion problem
  14. Beam lateral vibration problem
  15. Numerical simulations
  16. ...and 1 more sections

Key Result

Theorem 1

($l_\infty$ tomographykerenidis_quantum_2019). Given access to unitary $U$ such that $U|0\rangle=|x\rangle$ and its controlled version in time $T(U)$, there is a tomography algorithm with time complexity $O(T(U)\frac{\log N}{\epsilon_s^2})$ that produces unit vector $\widetilde{X}\in R^N$ such that

Figures (4)

  • Figure 1: Overall block of quantum Newton's method.
  • Figure 2: Structure of a 4-dimensional $M_F$.
  • Figure 3: Numerical Results. Panels (a),(b) represent the influence of sample error $\epsilon_s$ on the convergence process of nonlinear first-order diffusion problem and beam lateral vibration problem respectively.
  • Figure 4: Relationship between $\epsilon_s$ and convergent $||\bm F(\bm x)||$. Panels (a),(b) represent nonlinear first-order diffusion problem and beam lateral vibration problem respectively.

Theorems & Definitions (1)

  • Theorem 1