The finite element method (FEM) is a cornerstone numerical technique for solving partial differential equations (PDEs). Here, we present $\textbf{Qu-FEM}$, a fault-tolerant era quantum algorithm for the finite element method. In contrast to other quantum PDE solvers, Qu-FEM preserves the geometric flexibility of FEM by introducing two new primitives, the unit of interaction and the local-to-global indicator matrix, which enable the assembly of global finite element arrays with a constant-size linear combination of unitaries. We study the modified Poisson equation as an elliptic problem of interest, and provide explicit circuits for Qu-FEM in Cartesian domains. For problems with constant coefficients, our algorithm admits block-encodings of global arrays that require only $\tilde{\mathcal{O}}\left(d^2 p^2 n\right)$ Clifford+$T$ gates for $d$-dimensional, order-$p$ tensor product elements on grids with $2^n$ degrees of freedom in each dimension, where $n$ is the number of qubits representing the $N=2^n$ discrete grid points. For problems with spatially varying coefficients, we perform numerical integration directly on the quantum computer to assemble global arrays and force vectors. Dirichlet boundary conditions are enforced via the method of Lagrange multipliers, eliminating the need to modify the block-encodings that emerge from the assembly procedure. This work presents a framework for extending the geometric flexibility of quantum PDE solvers while preserving the possibility of a quantum advantage.

翻译：有限元方法（FEM）是求解偏微分方程（PDE）的一项基础性数值技术。本文提出 $\textbf{Qu-FEM}$，一种适用于有限元方法的容错时代量子算法。与其他量子偏微分方程求解器不同，Qu-FEM通过引入两个新的基本单元——相互作用单元和局部到全局指示矩阵——保持了有限元方法的几何灵活性，从而能够通过常数规模的酉算子线性组合来组装全局有限元数组。我们以修正的泊松方程作为感兴趣的椭圆问题进行研究，并为笛卡尔域中的Qu-FEM提供了明确的量子电路。对于常系数问题，我们的算法允许对全局数组进行块编码，该编码在每维具有 $2^n$ 个自由度的网格上，对于 $d$ 维、$p$ 阶张量积单元，仅需 $\tilde{\mathcal{O}}\left(d^2 p^2 n\right)$ 个 Clifford+$T$ 门，其中 $n$ 是表示 $N=2^n$ 个离散网格点的量子比特数。对于具有空间变化系数的问题，我们直接在量子计算机上进行数值积分以组装全局数组和力向量。狄利克雷边界条件通过拉格朗日乘子法施加，从而无需修改组装过程中产生的块编码。这项工作提出了一个框架，旨在扩展量子偏微分方程求解器的几何灵活性，同时保留实现量子优势的可能性。