发表机构
College of Mathematical Sciences, Harbin Engineering University（哈尔滨工程大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对全局VQA在空间异质PDE上的低效，提出分块VQA框架，通过动态分配网格与拟设、界面罚耦合及自适应重分块，降低误差和电路宽度，实现近端量子设备上的高保真求解。
AI 中文摘要
求解偏微分方程（PDE）的全局变分量子算法（VQA）框架通常依赖于均匀网格上的单一高表达力拟设，当解表现出空间异质复杂性（如局部奇异性或薄边界层）时，这种结构在效率上变得低下。局部非光滑特征会降低整个全局量子表示的收敛性，造成不必要的电路深度，并加剧贫瘠高原中梯度消失的风险。为克服此局限，我们提出一种针对解具有空间异质复杂性的PDE的分块VQA框架。计算域被分解为局部表示的量子子问题，其中网格预算和拟设族根据可计算的难度指标动态分配。人工块界面通过状态值、导数和物理通量的统一跳跃罚项耦合，而自适应重分块跟踪移动的粗糙区域，以在不过度增加全局量子比特开销的情况下保持局部精度。误差分析严格分离了空间离散、拟设表达力、优化收敛、有限采样噪声、重分块传递误差以及界面耦合贡献。在代表性椭圆、对流-扩散和Burgers问题上的可复现残差基模拟表明，与全局VQA方法相比，该方法实现了更低的近似误差和更小的峰值局部电路宽度。这些结果证实了分块量子资源局部化的有效性，表明在近期量子设备上无需单一全局高表达力拟设即可实现高保真PDE解。
英文摘要
Global variational quantum algorithm (VQA) frameworks for solving partial differential equations (PDEs) often rely on a single expressive ansatz over a uniform grid, which becomes structurally inefficient when solutions exhibit spatially heterogeneous complexity such as localized singularities or thin boundary layers. A localized nonsmooth feature can degrade the convergence of the entire global quantum representation, imposing unnecessary circuit depth and amplifying the risk of vanishing gradients in barren plateaus. To overcome this limitation, we propose a block-wise VQA framework for PDEs characterized by spatially heterogeneous solution complexity. The computational domain is decomposed into locally represented quantum subproblems, where grid budgets and ansatz families are dynamically assigned based on a computable difficulty indicator. Artificial block interfaces are coupled through unified jump penalties for state values, derivatives, and physical fluxes, while adaptive reblocking tracks moving rough regions to maintain local accuracy without excessive global qubit overhead. The error analysis rigorously separates spatial discretization, ansatz expressivity, optimization convergence, finite-shot sampling noise, transfer errors from reblocking, and interface-coupling contributions. Reproducible residual-based simulations on representative elliptic, advection--diffusion, and Burgers problems demonstrate lower approximation errors and reduced peak local circuit width compared to global VQA approaches. These results substantiate block-wise quantum resource localization, confirming that high-fidelity PDE solutions can be achieved on near-term quantum devices without requiring a single globally expressive ansatz.
Comments25 pages, 5 figures, 3 tables
