用量子算法加速求解微分方程，无需大量训练数据
Neural Quantum Spectral Operator Learning for Solving Partial Differential Equations
- 结合量子计算与神经网络，以弱形式框架实现无监督学习
- 在1D和2D问题上精度优于经典方法，且支持多种边界条件
- 适合对高效求解偏微分方程有需求的物理模拟研究者
偏微分方程（PDEs）是建模物理与工程系统的核心工具，但参数化PDE的重复求解仍具高计算成本。算子学习可实现快速代理推断，但通常依赖大量由高保真求解器生成的输入-输出配对数据。无监督算子学习框架虽减少数据依赖，却受限于计算瓶颈。为此，我们提出首个混合量子-经典算子学习框架：神经变分量子线性求解器（NVQLS），利用勒让德-加莱尔金弱形式。我们关键解决了VQLS能量最小化中的符号模糊性问题，防止错误解表示。同时引入神经嵌入，一种新型编码方案，将变化的外力与PDE系数映射为参数化量子电路表示。这些结构创新在高效态制备条件下具备理论计算复杂度优势，且相比代表性经典基线实现更优精度。在不同边界条件下的一维与二维参数化PDE验证中，NVQLS展现同时处理多样化输入的能力，提供可扩展的量子增强算子学习方案。
原文摘要 · Abstract (English)
Partial differential equations (PDEs) are central to modeling physical and engineering systems, but repeatedly solving parametric PDEs remains computationally expensive. Operator learning enables fast surrogate inference, yet typically requires large input-output paired datasets generated by costly high-fidelity PDE solvers. Unsupervised operator learning frameworks alleviate data dependency but remain hindered by computational bottlenecks. To address this, we propose Neural Variational Quantum Linear Solver (NVQLS), the first hybrid quantum-classical operator learning framework leveraging the Legendre--Galerkin weak formulation. We critically resolve the sign ambiguity in VQLS energy minimization, preventing erroneous solution representations. Additionally, we introduce a neural embedding, a novel encoding scheme to map varying forcings and PDE coefficients into parameterized quantum circuit representations. These structural innovations provide theoretical computational complexity advantages under efficient state preparation schemes, while achieving superior accuracy compared to a representative classical baseline. Validations on 1D and 2D parametric PDEs under diverse boundary conditions demonstrate NVQLS's capability to simultaneously process varying inputs, offering a scalable unsupervised approach to quantum-enhanced operator learning.
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