[论文解读] Solving Nonlinear Partial Differential Equations via a Hybrid Newton Method Using Quantum Linear System Solver

Maximilian Mandelt Buxadé, Stefan Langer+1 more|arXiv (Cornell University)|Mar 24, 2026

Quantum Computing Algorithms and Architecture被引用 0

一句话总结

论文引入了一种混合量子-经典牛顿方法，利用量子线性系统求解器（QLSS）近似求解内部线性系统，从而在CFD场景中具有潜在加速的非线性偏微分方程求解能力。

ABSTRACT

To approximate solutions of complex nonlinear partial differential equations remains a computational challenge, especially for sets of equations relevant in industry, such as Euler or Navier-Stokes equations. Even the most sophisticated computational fluid dynamic algorithms coupled with powerful supercomputers can not find approximate solutions for several design challenges in both adequate time and scale-resolving accuracy. One difficulty arises from solving high dimensional, strongly nonlinear partial differential equations, such as the Navier-Stokes equations, which capture the underlying physics. For nearly all classical algorithms, methods closely related to Newton's method are used to approximate a solution to the problem. Approximately solving the large-scale linear systems of equations occurring in this iterative scheme is generally a main contributor to the total computational complexity. In this paper a new quantum linear system solver supporting Newton's classical method to solve nonlinear partial differential equations is introduced. We present a new variant of the HHL algorithm, requiring less apriori information regarding the eigenvalues of the corresponding matrix. We apply this quantum linear system solver in a hybrid quantum-classical fashion to solve nonlinear partial differential equations. Moreover, a resource estimation for advanced use-cases of practical relevance is provided. Our results demonstrate how quantum computation may improve existing classical methodologies for solving nonlinear partial differential equations. This approach provides another promising application of quantum computers and presents a possible way forward for handling nonlinearities on inherently linear quantum systems.

研究动机与目标

• 为CFD与产业规模问题中的非线性PDE求解需求提供改进方案的动机。
• 提出一种混合牛顿方法，利用量子线性系统求解器来求解内部线性系统。
• 开发并分析一种基于HHL的QLSS变体，该变体需要较少的先验特征值信息。
• 给出数值模拟和资源估计，以评估潜在加速和可行性。

提出的方法

• 将牛顿法用于非线性PDE与量子线性系统求解器耦合，以近似内部线性求解。
• 引入一种使用黑箱态制备启发技术的HHL算法变体，用于在较少的特征值假设下编码倒数特征值。
• 利用量子相位估计和改良幅度编码来获得倒数特征值并执行后选择。
• 实现一个经典版本（模型QLSS）以在没有量子硬件时研究尺度性和误差行为。
• 在线性和非线性PDE（包括泊松方程和 Burgers 方程）上进行仿真，以评估收敛性和精度。

实验结果

研究问题

• RQ1 QLSS 支持的牛顿迭代是否能够对泊松和 Burgers 方程等非线性PDE 收敛？
• RQ2 在PDE的牛顿方法中使用QLSS的精度、资源和尺度化意味着什么？
• RQ3 在内部牛顿步骤的背景下，相较于经典线性求解器，所提QLSS变体有何差异？
• RQ4 在CFD相关问题中实现该混合方法的实际考虑因素与限制是什么？

主要发现

• 在仿真实验中，带有QLSS的混合牛顿方法在非线性泊松和 Burgers 方程上表现出收敛性。
• 特征值近似和逆的误差通过牛顿迭代传播，较高的QPE量子比特数可提高精度。
• 增加QPE量子比特数（m）可实现更快的残差收敛，在某些区域可在牛顿步中超过固定的Gauss-Seidel迭代。
• 对于固定的m，增大问题规模N需要更多迭代，但在同等努力下仍可达到残差，表明相较时间步进方案具有潜在的可扩展性优势。
• 研究提供了详细的资源讨论，并表明在量子硬件上对CFD类非线性PDE存在潜在加速路径。