We investigate time complexities of finite difference methods for solving the high-dimensional linear heat equation, the high-dimensional linear hyperbolic equation and the multiscale hyperbolic heat system with quantum algorithms (hence referred to as the "quantum difference methods"). For the heat and linear hyperbolic equations we study the impact of explicit and implicit time discretizations on quantum advantages over the classical difference method. For the multiscale problem, we find the time complexity of both the classical treatment and quantum treatment for the explicit scheme scales as $\mathcal{O}(1/\varepsilon)$, where $\varepsilon$ is the scaling parameter, while the scaling for the multiscale Asymptotic-Preserving (AP) schemes does not depend on $\varepsilon$. This indicates that it is still of great importance to develop AP schemes for multiscale problems in quantum computing.
翻译:我们调查了解决高维线性热方程式、高维线性双曲方程式和带有量子算法(称为“量子差异方法”)的多尺度双曲热系统等有限差异方法的时间复杂性。对于热和线性双曲方程式,我们研究了明确和隐含的时间分解对量子优于传统差异方法的影响。对于多尺度问题,我们发现对显性方案规模的古典处理和量子处理的时间复杂性为$\mathcal{O}(1/\varepsilon)$,其中, $\varepsilon$是缩放参数, 而多尺度的Asymptic-Prestive(AP) 计划的规模并不取决于$\vareplon(AP) 。这说明为量子计算中的多尺度问题制定AP计划仍然非常重要。