用于基于偏微分方程（PDE）的贝叶斯反问题的量子辅助框架
A quantum-assisted framework for PDE-based Bayesian inverse problems
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中文总结 AI 辅助
针对基于PDE的贝叶斯反问题，提出量子-经典混合框架，用量子处理器处理PDE演化与损失评估、经典计算机调整超参数，经实验验证其在采样噪声下的参数反演可行性，为相关问题提供量子辅助方案。
中文摘要 AI 辅助
量子计算为求解偏微分方程（PDE）提供了潜在优势，但现有大多数量子PDE求解器主要聚焦于制备对应解的量子态，而从这些量子态中高效恢复经典信息的研究仍较少。受限于读出环节，我们提出了一种用于贝叶斯PDE反问题的量子-经典混合框架：量子处理器演化PDE并评估带采样噪声的损失函数，经典计算机则调整高斯过程回归中的超参数，以探索下一个待测试候选解。为适配针对线性和半线性自治演化PDE的量子求解器，我们建议采用归一化量子态损失作为数据拟合函数，并结合量子PDE求解器与哈达玛测试来评估新的拟合函数，从而仅用有限数量的量子态副本即可提取有用的经典信息，无需重构完整的解向量。对误差传播和指定精度下损失评估的整体复杂度分析表明，在量子测量条件下，该新型数据拟合函数优于传统的L2损失。对一维和二维线性对流扩散方程的量子电路模拟（在近似和有限采样损失评估下），以及对非线性受迫粘性伯格斯方程的经典数值实验，均证明了所提方法在损失评估受采样噪声影响时仍能用于参数反演的可行性。该框架或可为基于PDE的反问题提供可行的量子辅助方案，并阐明量子PDE算法在解决完整量子到末端优化栈方面的潜力。
英文摘要
Quantum computing offers potential advantages for solving partial differential equations (PDEs). However, most existing quantum PDE solvers primarily focus on preparing quantum states for solutions, while the efficient recovery of classical information from these states remains less explored. Motivated by the readout limitation, we propose a quantum-classical hybrid framework for Bayesian PDE inversion problems: The quantum processor evolves the PDE and evaluate the loss function with sampling noises, while the classical computer tunes the hyper-parameters in the Gaussian Process Regression to explore the next trial candidate. To match the quantum solvers for linear and semi-linear autonomous evolution PDEs, we suggest to use a normalized quantum-state loss as the data-misfit function and evaluate the new misfit by combining quantum PDE solvers with the Hadamard test, thereby allowing us to extract useful classical information using only a limited number of quantum state copies without reconstructing the full solution vector. The analysis of error propagation and overall complexity of loss evaluation under a prescribed accuracy shows that the new data-misfit function outperforms the conventional L2-loss under quantum measurements. Quantum circuit simulations of 1D and 2D linear convection diffusion equations under approximate and finite sampling loss evaluations, together with classical numerical experiments on a nonlinear forced viscous Burgers equation, demonstrate the feasibility of the proposed approach for parameter inversion even when the loss evaluations are affected by sampling noise. This framework may provide a viable quantum-assisted scheme for PDE-based inverse problems and elucidate the potential of quantum PDE algorithms in addressing a complete quantum-to-end optimization stack.
