Quantum Block Encodings for Periodic Two-Phase Finite Element Operators: 2D Poisson and 2D Elasticity

2026-09-15

AI Summary

本文针对二维泊松和弹性周期有限元算子，提出了一种量子块编码方法，通过线性组合酉矩阵来实现精确编码，以期在量子算法中获得加速。

Abstract

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

Problem

• Quantum Block Encodings
• Two-Phase Finite Element Operators
• 2D Poisson
• 2D Elasticity
• Periodic

Innovation

• Quantum Block Encodings
• Two-Phase Finite Element Operators
• Shift Decomposition
• Linear Combination of Unitaries (LCU)
• Node-to-Element Oracle