Quantum computing offers a promising avenue for advancing computational methods in science and engineering. In this work, we introduce the quantum asymptotic numerical method (qANM), a framework for solving nonlinear problems using quantum computing. Based on the principle of high-order perturbation techniques, the proposed method uses Taylor series expansions to transform complex nonlinear systems into sequences of linear equations. We integrate the method with the variational quantum linear solver and a quantum-enhanced Jacobi method. Numerical simulations on a quantum simulator validate the convergence of the method. In particular, the high-order ANM formulation demonstrates robustness in addressing nonlinear problems by effectively capturing the solution path through Taylor series expansions. Furthermore, a highlight of this work is a proof-of-principle experiment on a superconducting quantum processor. Despite the noise inherent in near-term quantum hardware, the experiment achieves 98% accuracy in tracking the nonlinear solution path. We believe this work provides a useful reference for applying quantum computing to nonlinear computational mechanics.

翻译：量子计算为科学和工程领域的计算方法发展提供了有前景的途径。本文提出了一种基于渐近数值法的量子框架（qANM），用于利用量子计算求解非线性问题。该方法基于高阶摄动技术原理，通过泰勒级数展开将复杂非线性系统转化为一系列线性方程。我们将其与变分量子线性求解器以及量子增强雅可比方法相结合。在量子模拟器上进行的数值模拟验证了该方法的收敛性。特别地，高阶ANM公式通过泰勒级数展开有效捕捉解路径，展现了处理非线性问题的鲁棒性。此外，本文的亮点在于在超导量子处理器上进行了原理验证实验。尽管近期量子硬件存在固有噪声，该实验在追踪非线性解路径时仍达到了98%的准确率。我们相信这项工作为量子计算在非线性计算力学中的应用提供了有益参考。