Quantum computing offers a promising new avenue for advancing computational methods in science and engineering. In this work, we introduce the quantum asymptotic numerical method, a novel quantum nonlinear solver that combines Taylor series expansions with quantum linear solvers to efficiently address nonlinear problems. By linearizing nonlinear problems using the Taylor series, the method transforms them into sequences of linear equations solvable by quantum algorithms, thus extending the convergence region for solutions and simultaneously leveraging quantum computational advantages. Numerical tests on the quantum simulator Qiskit confirm the convergence and accuracy of the method in solving nonlinear problems. Additionally, we apply the proposed method to a beam buckling problem, demonstrating its robustness in handling strongly nonlinear problems and its potential advantages in quantum resource requirements. Furthermore, we perform experiments on a superconducting quantum processor from Quafu, successfully achieving up to 98% accuracy in the obtained nonlinear solution path. We believe this work contributes to the utility of quantum computing in scientific computing applications.

翻译：量子计算为科学与工程领域的计算方法发展提供了一条前景广阔的新途径。本文提出量子渐近数值方法，这是一种结合泰勒级数展开与量子线性求解器的新型量子非线性求解器，能够高效处理非线性问题。该方法通过泰勒级数对非线性问题进行线性化处理，将其转化为可由量子算法求解的线性方程组序列，从而扩展了解的收敛区域，同时充分发挥量子计算的优势。在量子模拟器Qiskit上的数值测试验证了该方法求解非线性问题的收敛性与准确性。此外，我们将所提方法应用于梁屈曲问题，展示了其在处理强非线性问题时的鲁棒性及在量子资源需求方面的潜在优势。进一步地，我们在Quafu超导量子处理器上开展实验，成功获得精度高达98%的非线性解路径。我们相信这项工作有助于推动量子计算在科学计算应用中的实用化进程。