This manuscript presents the Quantum Finite Element Method (Q-FEM) developed for use in noisy intermediate-scale quantum (NISQ) computers, and employs the variational quantum linear solver (VQLS) algorithm. The proposed method leverages the classical FEM procedure to perform the unitary decomposition of the stiffness matrix and employs generator functions to design explicit quantum circuits corresponding to the unitaries. Q-FEM keeps the structure of the finite element discretization intact allowing for the use of variable element lengths and material coefficients in FEM discretization. The proposed method is tested on a steady-state heat equation discretized using linear and quadratic shape functions. Numerical verification studies demonstrate that Q-FEM is effective in converging to the correct solution for a variety of problems and model discretizations, including with different element lengths, variable coefficients, and different boundary conditions. The formalism developed herein is general and can be extended to problems with higher dimensions. However, numerical examples also demonstrate that the number of parameters for the variational ansatz scale exponentially with the number of qubits to increase the odds of convergence, and deterioration of system conditioning with problem size results in barren plateaus, and hence convergence difficulties.

翻译：本文提出了一种用于含噪声中等规模量子（NISQ）计算机的量子有限元方法（Q-FEM），并采用了变分量子线性求解器（VQLS）算法。所提出的方法利用经典FEM流程执行刚度矩阵的酉分解，并采用生成器函数来设计与酉矩阵对应的显式量子电路。Q-FEM保持了有限元离散化的结构不变，从而允许在FEM离散化中使用可变的单元长度和材料系数。该方法在使用线性和二次形函数离散化的稳态热传导方程上进行了测试。数值验证研究表明，对于包括不同单元长度、变系数及不同边界条件在内的各种问题和模型离散化，Q-FEM均能有效收敛至正确解。本文发展的形式体系具有通用性，可扩展至高维问题。然而，数值算例也表明，变分拟设的参数数量随量子比特数呈指数增长以提高收敛概率，且系统条件数随问题规模增大而恶化，导致出现贫瘠高原现象，从而引发收敛困难。