AI 中文总结
该研究提出基于离散拉普拉斯算子精确块编码的泊松方程变分量子线性求解器，仅需单电路完成代价评估，经三类物理基准测试验证性能，且发现经典优化器选择影响变分优化进展。
AI 中文摘要
我们提出了一种基于离散拉普拉斯算子精确块编码的泊松方程变分量子线性求解器（VQLS），并在物理驱动的基准测试中验证了其性能。与基于线性组合幺正变换（LCU）的VQLS不同，后者每次代价函数评估所需的不同电路数量为O(L²)，其中L是离散拉普拉斯算子的LCU分解项数，而本方法仅需单电路即可完成代价评估。我们进一步通过实验证明，经典优化器的选择会显著影响变分优化停止取得进展的位置。该求解器在三类问题上进行了基准测试：带有正弦驱动项且满足狄利克雷边界条件的泊松方程、带有局域高斯源且满足狄利克雷边界条件的稳态热传导问题，以及二维顶盖驱动空腔流动的压力泊松方程——在该问题中，求解器在满足诺伊曼边界条件下每时间步调用一次。
英文摘要
We present a variational quantum linear solver (VQLS) for the Poisson equation built on an exact block encoding of the discrete Laplacian, and demonstrate its performance on physically motivated benchmarks. Unlike LCU-based VQLS where the number of distinct circuits required per cost-function evaluation is $\mathcal{O}(L^2)$, where $L$ is the number of terms in the LCU decomposition of the discrete Laplacian operator, this approach requires only a single circuit for cost evaluation. We further empirically demonstrate that the choice of classical optimizer materially affects where the variational optimization ceases to make progress. The solver is benchmarked on three problems: a Poisson equation with sinusoidal forcing and a steady-state heat conduction problem with a localized Gaussian source, both with Dirichlet boundaries, and the pressure-Poisson equation of a two-dimensional lid-driven cavity flow, in which the solver is invoked once per time step under Neumann boundary conditions.
Comments14 pages, 11 figures
