用量子卷积网络求解偏微分方程，兼顾精度与量子电路效率。
Quantum Neural Physics: Solving Partial Differential Equations on Quantum Simulators using Quantum Convolutional Neural Networks
- 将偏微分方程的离散算子转为量子卷积原语，嵌入经典多层求解框架。
- 在理想并行模型下，量子电路深度仅随区块大小对数增长（O(log K)）。
- 适合关注量子算法结构兼容性与数值稳定性的科学计算研究者。
神经物理将偏微分方程（PDE）的局部离散化重构为固定卷积算子，为科学机器学习中的数据驱动代理建模提供保持物理规律的替代方案。然而现有实现仍局限于经典硬件，未直接对接量子算子设计。为此，我们提出「量子神经物理」框架，并开发混合量子-经典卷积神经网络多网格求解器（HQC-CNNMG）。该方法将解析给定的模板算子映射为局部量子卷积原语，嵌入经典多层W循环架构中，结合科学机器学习的算子视角与多网格求解的数值严谨性。利用振幅编码、线性酉组合（LCU）及量子傅里叶变换（QFT），所得局部量子算子可实现对数深度实现，电路深度在理想并行模型下对编码块大小K呈O(log K)增长。在泊松方程、瞬态扩散、对流-扩散及不可压缩纳维-斯托克斯问题上的数值实验表明，该方法具数值一致性、稳定的多层行为，且在无噪声模拟器上具备工作流可行性。与代表性量子线性求解范式对比显示，其核心优势在于局部电路深度、数值鲁棒性与PDE结构兼容性的平衡，而非全量子全局求逆。
原文摘要 · Abstract (English)
Neural Physics recasts local discretisations of partial differential equations (PDEs) as fixed convolutional operators, providing a physics-preserving alternative to data-driven surrogate modelling in scientific machine learning. However, existing realizations remain largely confined to classical AI hardware and do not directly connect to quantum structured operator design. To bridge this gap, we introduce a \emph{Quantum Neural Physics} framework and develop a Hybrid Quantum-Classical CNN Multigrid Solver (HQC-CNNMG). The proposed method maps analytically prescribed stencil operators to local quantum convolutional primitives and embeds them within a classical multilevel W-cycle architecture, combining the operator-centric view of scientific ML with the numerical rigor of multigrid solvers. Using amplitude encoding together with the Linear Combination of Unitaries (LCU) and the Quantum Fourier Transform (QFT), the resulting local quantum operators admit logarithmic-depth implementation, with circuit depth scaling as $\mathcal{O}(\log K)$ for an encoded block of size $K$ under the idealized parallel circuit model considered here. Numerical experiments on Poisson, transient diffusion, convection--diffusion, and incompressible Navier--Stokes problems demonstrate numerical consistency, stable multilevel behaviour, and workflow-level feasibility on noiseless simulators. Comparisons with representative quantum linear solver paradigms further show that the main strength of HQC-CNNMG lies in its balanced trade-off among local circuit depth, numerical robustness, and compatibility with PDE structure, rather than in fully quantum global inversion.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。