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

TL;DR

This paper introduces a quantum Newton's method (QNM) for solving nonlinear algebraic systems, leveraging quantum linear solvers and data structures to achieve sublinear complexity, potentially surpassing classical algorithms.

Contribution

The paper develops a quantum Newton's method that integrates quantum linear solvers and data conversion techniques, enabling efficient solutions to nonlinear systems with quantum advantage.

Findings

QNM complexity per iteration is O(log^4 N / ε_s^2).

QNM remains effective when ε_s >> 1/√N.

QNM offers sublinear complexity in N, surpassing classical methods.

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 $\epsilon_s$ 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 $O(\log^4N/\epsilon_s^2)$. Through numerical simulation, we find that when $\epsilon_s>>1/\sqrt{N}$, QNM is still effective, so the complexity of QNM is sublinear with $N$, which provides quantum advantage compared with the optimal classical algorithm.