发表机构
School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对反应扩散问题的Q1有限元离散，提出结构感知的量子BPX预条件子构造，利用局部L2正交基表示状态，实现高效量子态制备，误差可控，查询复杂度仅多对数依赖于网格细化。
AI 中文摘要
我们提出了一种结构感知的量子BPX预条件子构造方法，用于单位立方体上常系数反应-扩散方程（包括泊松方程）的Q1有限元离散。基于现有的量子有限元和有限差分方法，我们通过显式的一维映射、张量积和保范的层间传递来表达预条件算子。对称预条件产生多水平系数，但其良好的条件数本身并不能确保在原始有限元节点基下高效恢复状态。我们转而使用构造中基础的局部$L^2$正交基作为同一有限元函数的输出表示。其系数范数等于连续$L^2$范数。假设可直接访问预条件右端项，我们准备此正交系数态，其表示函数误差至多为$\varepsilon$。期望查询复杂度为\\[ \mathcal{O}\\!\Big(dL^3\frac{\\|f\\|_{L^2}}{\\|u\\|_{L^2}} \log\\!\Big(2+\frac{L\\|f\\|_{L^2}}{\varepsilon}\Big)\Big), \\] 其中$u$为精确解，$f$为右端项，$d$为空间维度，$L=\log_2(1/h)$，$h$为网格宽度。该界在$0<\varepsilon\le\\|u\\|_{L^2}/2$时成立，并且对于固定连续数据，仅依赖于网格细化的多对数。我们还分析了线性泛函估计，并通过一维UnitaryLab实验展示了该构造和连续误差界。
英文摘要
We present a structure-aware construction of quantum BPX preconditioners for Q1 finite element discretizations of constant-coefficient reaction--diffusion equations on the unit cube, including the Poisson equation. Building on existing quantum finite element and finite difference methods, we express the preconditioned operator through explicit one-dimensional maps, tensor products, and norm-preserving interlevel transfers. Symmetric preconditioning gives multilevel coefficients, and its favorable conditioning alone does not ensure efficient recovery of a state in the original finite element nodal basis. We instead use the local $L^2$-orthonormal basis underlying the construction as the output representation of the same finite element function. Its coefficient norm equals the continuous $L^2$ norm. Assuming direct access to the preconditioned right-hand side, we prepare this orthonormal coefficient state with a represented function error at most $\varepsilon$. The expected query complexity is \[ \mathcal{O}\!\Big(dL^3\frac{\|f\|_{L^2}}{\|u\|_{L^2}} \log\!\Big(2+\frac{L\|f\|_{L^2}}{\varepsilon}\Big)\Big), \] where $u$ is the exact solution, $f$ is the right-hand side, $d$ is the spatial dimension, and $L=\log_2(1/h)$ for mesh width $h$. The bound holds for $0<\varepsilon\le\|u\|_{L^2}/2$ and depends only polylogarithmically on mesh refinement for fixed continuous data. We also analyze linear-functional estimation and illustrate the construction and continuous error bounds with a one-dimensional UnitaryLab experiment.
Commentsquantum BPX preconditioning
