While quantum computing provides an exponential advantage in solving the system of linear equations, there is little work to solve the 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∞ tomography with sample error ϵs 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(log4N∕ϵs2). Through numerical simulation, we find that when ϵs≫1∕N, QNM is still effective, so the complexity of QNM is sublinear with N, which provides quantum advantage compared with the optimal classical algorithm.

中文翻译：

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求解非线性方程组的量子牛顿法

虽然量子计算在求解线性方程组方面提供了指数优势，但用量子计算求解非线性方程组的工作却很少。我们提出了量子牛顿法 (QNM) 来求解ñ基于牛顿法的非线性方程组。在 QNM 中，我们使用量子线性系统求解器在牛顿法的每次迭代中求解线性方程组。我们使用特定的量子数据结构和l∞有样本错误的断层扫描εs实现QNM两次迭代之间的经典-量子数据转换过程，从而构建QNM的全过程。QNM在每次迭代中的复杂度为○(日志4ñ∕εs2). 通过数值模拟，我们发现当εs≫1∕ñ, QNM 仍然有效，所以 QNM 的复杂度是次线性的ñ，与最优经典算法相比，它提供了量子优势。