周期有限差分算子的矩结构块编码
Moment-Structured Block Encodings of Periodic Finite-Difference Operators
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中文总结 AI 辅助
研究周期网格上平移不变有限差分算子的块编码，通过矩阶确定相关性质并导出最优性准则，该框架包含拉普拉斯算子最优构造，可验证新实例，还导出成功概率下限及平流扩散族块编码常数。
中文摘要 AI 辅助
块编码是量子线性代数算法中访问矩阵数据的标准技术，其实现直接影响次归一化，进而控制算法的成功概率、模拟时间和下游成本。目前只有少数算子能显式构造出具有可证明最优次归一化的块编码。本文开发了一个框架，用于对周期网格上的平移不变有限差分算子进行块编码，这些算子是科学计算核心的常系数偏微分方程的有限差分离散化。我们表明，这些模板的矩阶可用于同时确定近似的连续算子、傅里叶符号的消失阶以及块编码的成本。由此，我们导出了一个作为模板系数函数的闭式最优性准则，可验证构造是否在整个算子族中达到最优次归一化，并量化未达到时的差距。该框架包含了文献中拉普拉斯算子的最优构造，可用于验证更高偶数阶的新实例，包括双调和算子。此外，我们导出了由算子符号的谱特性参数化的成功概率下限，并为平流扩散族的块编码找到了明确的常数，此前该族不存在明确的空间块编码。
英文摘要
Block encoding is the standard technique for accessing matrix data in quantum linear-algebra algorithms. Its implementation directly affects its subnormalization, which in turn controls the algorithm's success probability, simulation time, and downstream costs. Explicit construction of block encodings with provably optimal subnormalization exists for only a handful of operators, with bespoke calculations used in its design. In this work, we develop a framework to block encode translation-invariant finite-difference operators on a periodic grid. These operators are the finite-difference discretizations of the constant-coefficient partial differential equations that sit at the core of scientific computing. We show that the moment order of these stencils can be used to simultaneously determine the continuum operator approximated, the vanishing order of the Fourier symbol, and the cost of the block encoding. From there, we derive a closed-form optimality criterion as a function of the stencil coefficients, which certifies whether the construction attains the optimal subnormalization for an entire operator family, uniformly in grid size, and quantifies the gap when it does not. The framework subsumes optimal constructions for the Laplacian operator in the literature and can be used to certify new instances at higher even orders, including the biharmonic operator. Furthermore, we derive success-probability floors parameterized by spectral properties of the operator's symbol and find explicit constants for the block encoding of the advection-diffusion family for which no prior explicit spatial block encoding exists.
