发表机构
University of Cambridge（剑桥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种量子有限元方法（IQFEM），用于浸没边界非均匀泊松问题，通过LCU和QSVT实现多对数门复杂度，数值验证线性或二次收敛。
AI 中文摘要
量子计算有望以比经典计算更有利的复杂度规模来解决计算力学问题。为实现这一潜力，有限元方法的量子兼容表述至关重要。我们针对一般域上由浸没均匀笛卡尔网格离散化的、具有空间变化系数的泊松问题，提出了这样一种表述。所得的带状稀疏系统矩阵被编码为对角矩阵（其元素为多项式）与其与移位矩阵乘积之和。矩阵之和使用线性组合酉算子（LCU）技术形成。所得的块编码系统矩阵可以使用多对数数量的基本门来实现。对于浸没边界，块编码系统矩阵进一步使用一个由零和一组成的对角指示矩阵进行处理。该指示矩阵由一个整数值水平集函数定义，该函数通过布尔集合运算由简单几何基元构成。所需的整数运算使用量子算术即时计算。所得的块编码线性方程组使用量子奇异值变换（QSVT）求解。包括QSVT在内的总门数为$O(N_{\text{tot}}^{2/d} \operatorname{polylog}(N_{\text{tot}}))$，其中$d$是问题维度，$N_{\text{tot}}$是网格点数。在确认理论复杂度估计的数值示例中，对于浸没问题，解近似线性收敛，对于单连通域，解二次收敛。
英文摘要
Quantum computing offers the potential to solve computational mechanics problems with a more favourable complexity scaling than classically possible. To realise this potential, a quantum-compatible formulation of the finite element method is essential. We present such a formulation for Poisson problems with spatially varying coefficients on general domains discretised by an immersed uniform Cartesian grid. The resulting banded sparse system matrices are encoded as the sum of diagonal matrices with polynomial entries and their products with shift matrices. The sum of the matrices is formed using the linear combination of unitaries (LCU) technique. The resulting block-encoded system matrix can be implemented using a polylogarithmic number of elementary gates. For immersed boundaries, the block-encoded system matrix is further processed using a diagonal indicator matrix with zeros and ones. The indicator matrix is defined in terms of an integer-valued level-set function composed of simple geometric primitives via Boolean set operations. The required integer operations are computed on the fly using quantum arithmetic. The resulting block-encoded linear system of equations is solved using quantum singular value transformation (QSVT). The overall gate count, including the QSVT, is $O(N_{\text{tot}}^{2/d} \operatorname{polylog}(N_{\text{tot}}))$, where $d$ is the problem dimension and $N_{\text{tot}}$ the number of grid points. In numerical examples that confirm the theoretical complexity estimates, the solutions converge approximately linearly for immersed problems and quadratically for simply connected domains.
