AI 中文总结
研究二阶边值问题，通过引入框架构造连续微分算子有限维对应物，利用外导数等导出离散实现并转化为星局部更新规则，得到简单可扩展量子电路，为偏微分方程量子算法提供系统途径。
AI 中文摘要
二阶边值问题是计算科学的核心，但基于标准矩阵的数值公式可能会掩盖量子电路可利用的局部几何结构。本文引入一个框架，以与量子计算兼容的形式显式构造连续微分算子的有限维对应物。从外导数、其伴随和霍奇算子出发，我们在原对偶胞腔复形上导出二阶算子的离散实现，并将其重新表述为星局部更新规则，通过显式函数返回相关边界链而非矩阵表示。这产生了简单、统一且可扩展的量子电路。我们展示了散度 - 梯度和旋度 - 旋度算子的构造，表明相同的星局部编译原理在通用框架内跨越不同算子、胞腔复形和流形维度。更一般地，该框架为偏微分方程的量子算法提供了一条系统途径。
英文摘要
Second-order boundary value problems are central to computational science, yet standard matrix-based numerical formulations can obscure the local geometric structure that quantum circuits may exploit. Here we introduce a framework for explicitly constructing finite-dimensional counterparts of continuous differential operators in a form compatible with quantum computation. Starting from the exterior derivative, its adjoint, and the Hodge operator, we derive discrete realizations of second-order operators on primal--dual cell complexes and reformulate them as star-local update rules expressed through explicit functions that return the relevant bounding chains rather than through matrix representations. This yields simple, uniform, and scalable quantum circuits. We demonstrate the construction for div--grad and curl--curl operators, showing that the same star-local compilation principle extends across different operators, cell complexes, and manifold dimensions within a common framework. More generally, the framework provides a systematic route to quantum algorithms for partial differential equations.
Commentssubmitted to Quantum Science and Technology - IOPscience
