发表机构
University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对二维周期两相有限元算子，提出精确块编码，通过节点到单元与材料预言机实现LCU分解，项数与网格无关，并给出闭式次归一化。
AI 中文摘要
量子算法需要使用块编码将线性算子嵌入酉算子中。这些块编码的成本决定了能否实现量子加速。针对二维均匀标量算子，已有高效的“移位分解”编码被提出。在此，我们为二维均匀弹性、二维两相双线性泊松以及二维两相弹性周期有限元算子给出了精确的块编码。对于二维均匀弹性，所得的酉算子线性组合（LCU）对每个泊松比ν和每个网格分辨率均包含L=17项，且次归一化具有闭式形式α=E(33+ν)/[6(1-ν²)]，其超过‖K‖₂的因子为(33+ν)/24，该因子与分辨率无关。为处理两相周期微结构，我们引入了两个角色不同的预言机：一个由循环移位构建的节点到单元预言机，将节点索引携带到其四个关联单元索引；另一个是微结构特定的材料预言机，用符号标记相成员关系。由此，两相泊松和弹性分别产生25项和57项的LCU；后者在ν=1/3时减少至49项。项数与网格分辨率、体积分数和相衬度无关。次归一化以闭式形式给出，且与网格分辨率和体积分数无关。在标量情形下，当节点完全位于较硬相时，可达到紧界α=‖K‖∞。在弹性情形下，该界无法达到，因为剪切耦合在行中引入了正负两种符号的条目。开源实现可在以下网址获取：此 https URL
英文摘要
Quantum algorithms require embedding linear operators into unitaries using block encodings. The costs associated with these block encodings determine whether a quantum speedup is achieved. Efficient \emph{shift decomposition} encodings have been proposed for 2D homogeneous scalar operators. Here, we present exact block encodings for 2D homogeneous elasticity, 2D two-phase bilinear Poisson, and 2D two-phase elasticity periodic finite-element operators. For 2D homogeneous elasticity, the resulting linear combination of unitaries (LCU) has $L = 17$ terms for every Poisson ratio $ν$ and every mesh resolution, and the subnormalization is the closed form $α= E(33+ν)/\bigl[6(1-ν^{2})\bigr]$, exceeding $\|\mathbf{K}\|_{2}$ by the resolution-independent factor $(33+ν)/24$. To address two-phase periodic microstructures, we introduce two oracles with distinct roles: a node-to-element oracle, built from cyclic shifts, that carries a nodal index to each of the four incident element indices, and a microstructure-specific material oracle that marks phase membership with a sign. This yields 25 and 57 LCU terms for two-phase Poisson and elasticity, respectively; the latter reduces to 49 at $ν= 1/3$. The term counts are independent of mesh resolution, volume fraction, and phase contrast. The subnormalizations, provided in closed form, are independent of mesh resolution and volume fraction. In the scalar case, the tight bound $α= \|\mathbf{K}\|_\infty$ is achieved whenever a node lies entirely in the stiffer phase. In the elasticity case, that bound is not attained, since the shear coupling contributes entries of both signs to a row. An open-source implementation is available at https://github.com/UW-ERSL/PyBlockEncode
