AI 中文总结
研究双调和边值问题高维离散化难题，通过构建针对不同边界条件的显式块编码开发量子线性系统算法，涵盖多种离散化方法及相关分析，经数值实验验证了离散化和线性求解的有效性。
AI 中文摘要
双调和方程是典型的四阶偏微分方程，其高维离散化存在自由度迅速增长和严重病态问题。我们通过构建针对周期、简支和狄利克雷 - 诺伊曼边界条件的显式块编码来开发QSVT - VTAA量子线性系统算法。对于周期和简支问题，傅里叶和正弦变换对角化产生具有二阶算子条件数缩放的增广泊松系统。对于狄利克雷 - 诺伊曼问题，我们引入二阶边界校正有限差分离散化，建立与网格无关的稳定性，并构造所得非对称矩阵的显式块编码。我们还制定了具有附加边界未知数的耦合拉普拉斯系统，并根据完整增广矩阵的条件数来表征其复杂性。分析涵盖离散化误差、块编码归一化、门复杂度以及在幅度输入和量子态输出模型下的解提取。数值实验验证了所提出的离散化和相应的线性求解。
英文摘要
The biharmonic equation is a prototypical fourth-order partial differential equation whose high-dimensional discretization suffers from rapidly growing degrees of freedom and severe ill-conditioning. We develop QSVT--VTAA quantum linear-system algorithms by constructing explicit block-encodings tailored to periodic, simply supported, and Dirichlet--Neumann boundary conditions. For periodic and simply supported problems, Fourier and sine-transform diagonalizations yield augmented Poisson systems with the condition-number scaling of a second-order operator. For Dirichlet--Neumann problems, we introduce a second-order boundary-corrected finite-difference discretization, establish mesh-independent stability, and construct an explicit block-encoding of the resulting nonsymmetric matrix. We also formulate a coupled-Laplace system with additional boundary unknowns and characterize its complexity in terms of the condition number of the complete augmented matrix. The analysis covers discretization error, block-encoding normalization, gate complexity, and solution extraction under an amplitude-input and quantum-state-output model. Numerical experiments validate the proposed discretizations and the corresponding linear solves.
Comments25 pages,16 figures
