对称正定系统通过量子本征值变换的量子梯度流算法：迈向量子计算机辅助工程
Quantum Gradient Flow Algorithm for Symmetric Positive Definite Systems via Quantum Eigenvalue Transformation: Towards Quantum CAE
AI总结：
本文提出量子梯度流算法，用于高效求解对称正定系统，通过变分原理实现快速收敛，展示了其在量子CAE中的应用潜力。
摘要：
本文提出了一种名为量子梯度流算法（QGFA）的新型量子算法，用于求解对称正定（SPD）线性系统，基于变分公式和时间演化动力学。传统量子线性求解器，如量子矩阵逆算法（QMIA），通过量子信号处理（QSP）来近似矩阵逆。然而，QMIA存在一个关键缺点：其计算效率随着条件数的增加而下降。相比之下，经典SPD线性求解器，如最速下降法和共轭梯度法，以其快速收敛而闻名，这源于SPD系统的变分优化原理。受此启发，我们开发了QGFA，通过相应二次能量泛函的梯度流过程获得解向量。为了验证所提出的方法，我们将其应用于位移基有限元法（FEM）来求解二维线弹性问题在平面应力条件下。该算法即使在使用少量QSP相位因子时也能准确收敛到经典FEM解。与QMIA相比，QGFA在合适初始状态初始化时实现了更低的相对误差和更快的收敛速度，展示了其作为高效预条件量子线性求解器的潜力。所提出的框架为经典迭代求解器与量子计算范式之间提供了物理可解释的连接。这些发现表明，QGFA可以成为未来量子计算机辅助工程（量子CAE）发展基础，包括非线性和多物理场模拟。
英文摘要：
In this study, we propose the Quantum Gradient Flow Algorithm (QGFA), a novel quantum algorithm for solving symmetric positive definite (SPD) linear systems based on the variational formulation and time-evolution dynamics. Conventional quantum linear solvers, such as the quantum matrix inverse algorithm (QMIA), focus on approximating the matrix inverse through quantum signal processing (QSP). However, QMIA suffers from a crucial drawback: its computational efficiency deteriorates as the condition number increases. In contrast, classical SPD linear solvers, such as the steepest descent and conjugate gradient methods, are known for their fast convergence, which stems from the variational optimization principle of SPD systems. Inspired by this, we develop QGFA, which obtains the solution vector through the gradient-flow process of the corresponding quadratic energy functional. To validate the proposed method, we apply QGFA to the displacement-based finite element method (FEM) for two-dimensional linear elastic problems under plane stress conditions. The algorithm demonstrates accurate convergence toward classical FEM solutions even with a moderate number of QSP phase factors. Compared with QMIA, QGFA achieves lower relative errors and faster convergence when initialized with suitable initial states, demonstrating its potential as an efficient preconditioned quantum linear solver. The proposed framework provides a physically interpretable connection between classical iterative solvers and quantum computational paradigms. These findings suggest that QGFA can serve as a foundation for future developments in Quantum Computer-Aided Engineering (Quantum CAE), including nonlinear and multiphysics simulations.
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