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While quantum computing provides an exponential advantage in solving systemof linear equations, there is little work to solve system of nonlinearequations with quantum computing. We propose quantum Newton's method (QNM) forsolving $N$-dimensional system of nonlinear equations based on Newton's method.In QNM, we solve the system of linear equations in each iteration of Newton'smethod with quantum linear system solver. We use a specific quantum datastructure and $l_{\infty}$ tomography with sample error $\epsilon_s$ toimplement the classical-quantum data conversion process between the twoiterations of QNM, thereby constructing the whole process of QNM. Thecomplexity of QNM in each iteration is $O(\log^4N/\epsilon_s^2)$. Throughnumerical simulation, we find that when $\epsilon_s>>1/\sqrt{N}$, QNM is stilleffective, so the complexity of QNM is sublinear with $N$, which providesquantum advantage compared with the optimal classical algorithm.

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