发表机构
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology（大连理工大学力学与航空航天学院工业装备结构分析优化及CAE软件国家重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出基于体素的块变分量子线性求解器，结合结构化分解、最小势能目标和批量量子测试，减少酉项与迭代次数，实现固体静力分析的高效混合量子-经典求解。
AI 中文摘要
在固体力学中，大规模静力问题的有限元离散化会产生大型稀疏线性系统，其求解需要大量的计算时间和内存。变分量子线性求解器（VQLS）提供了一条混合量子-经典的路径，但其在量子有限元分析中的应用受到非酉矩阵分解、贫瘠高原以及期望值测量成本的限制。我们提出了一种基于体素的块变分量子线性求解器（Voxel-BVQLS），它结合了结构化矩阵分解、最小势能原理和批量量子测试。首先，我们利用体素网格有限元矩阵的递归块带状结构，构建了刚度矩阵的LCU分解，其中酉项的数量与问题规模无关。其次，我们使用最小势能目标替代传统的VQLS损失函数来优化拟设参数，从而在所研究的问题中缓解贫瘠高原现象。第三，我们引入了一种块-哈达玛测试，其电路直接估计多个内积的加权和，减少了每次迭代所需的电路配置数量。我们使用三个示例在无噪声经典模拟中评估了所提出的方法。这些示例表明，该方法减少了LCU分解中的酉项数量以及收敛所需的迭代次数，同时仍能产生有限精度的解。因此，Voxel-BVQLS为规则网格上的量子有限元分析提供了一个结构化的混合量子-经典框架。
英文摘要
In solid mechanics, finite element discretization of large-scale static problems produces large sparse linear systems whose solution requires substantial computation time and memory. The variational quantum linear solver (VQLS) offers a hybrid quantum-classical route, but its use in quantum finite element analysis is limited by the decomposition of nonunitary matrices, barren plateaus, and the measurement cost of expectation values. We propose a voxel-based block variational quantum linear solver (Voxel-BVQLS) that combines structured matrix decomposition, the principle of minimum potential energy, and batched quantum tests. First, we construct an LCU decomposition of the stiffness matrix from the recursive block-banded structure of voxel-grid finite element matrices, with the number of unitary terms bounded independently of the problem size. Second, we optimize the ansatz parameters using a minimum-potential-energy objective in place of a conventional VQLS loss function, thereby mitigating barren plateaus in the studied problems. Third, we introduce a block-Hadamard test whose circuit directly estimates weighted sums of multiple inner products, reducing the number of circuit configurations required per iteration. We assessed the proposed method in noiseless classical simulations using three examples. These examples show that the method reduces both the number of unitary terms in the LCU decomposition and the number of iterations required to converge, while still yielding solutions of finite accuracy. Voxel-BVQLS thus provides a structured hybrid quantum-classical framework for quantum finite element analysis on regular grids.
Comments35 pages, 22 figures, 5 tables
