AI 中文总结
该研究针对近期量子设备求解微分方程的资源瓶颈，提出嵌入硬约束的PIQML框架，用参数化量子电路建模，以参数偏移规则算梯度，在多类方程上验证了其高精度求解非线性动力学的能力。
AI 中文摘要
基于线性系统方法的量子算法求解微分方程所需的量子比特与精度资源超出了近期设备的能力范围。为应对这些挑战，本研究提出了一种专为噪声中等规模量子（NISQ）时代设计的嵌入硬约束的物理信息量子机器学习（PIQML）框架。在该框架中，参数化量子电路充当机器学习模型，输入变量通过傅里叶特征映射编码至高维特征空间。随后，为消除关键物理条件下的近似误差，解由经严格设计的函数映射器构建，该映射器以硬约束形式解析地施加初始条件。重要的是，我们使用参数偏移规则计算关于输入变量的导数——这是一种量子原生梯度评估技术，可避免经典离散化。与针对抽象数据模式的通用损失函数不同，我们的损失函数聚焦于微分方程残差与参考数据。此设计确保训练后的模型不仅能近似数据，还能内在满足微分方程表达的物理约束。我们在包括高度振荡方程在内的多个微分方程上验证了该方法，证明其处理具挑战性非线性动力学的能力。结果表明，我们的量子模型成功学习到解，与高精度经典数值基准吻合度高。
英文摘要
Quantum algorithms based on linear-system approaches for solving differential equations demand qubit and precision resources beyond near-term capabilities. To address these challenges, this work proposes a physics-informed quantum machine learning (PIQML) framework with hard constraint embedding, specifically designed for NISQ era. Within this framework, parameterized quantum circuits serve as machine learning models, where the input variable is encoded into a high-dimensional feature space via a Fourier feature map. Subsequently, to eliminate approximation errors in critical physical conditions, the solution is constructed through a rigorously designed function mapper that analytically enforces initial conditions as hard constraints. Crucially, we compute derivatives with respect to the input variable using the parameter-shift rule---a quantum native gradient evaluation technique that avoids classical discretization. Unlike generic loss functions that target abstract data patterns, our loss function focuses on the differential equation residual and reference data. This design ensures that the trained model not only approximates the data but also intrinsically satisfies the physical constraint expressed by the DE itself. Our method is validated on several differential equations, including highly oscillatory ones, demonstrating its capability to tackle challenging nonlinear dynamics. Results demonstrate that our quantum model successfully learns the solution, showing close agreement with a high-precision classical numerical benchmark.
