论文标题
线性高维和多尺度偏微分方程的量子差异方法的时间复杂性分析
Time complexity analysis of quantum difference methods for linear high dimensional and multiscale partial differential equations
论文作者
论文摘要
我们研究了有限差异方法的时间复杂性，以求解具有量子算法的高维线性热方程，高维线性双曲方程和多尺度双曲热系统（因此称为“量子差异方法”）。对于热和线性双曲方程，我们研究了明确和隐式时间离散对量子优势比经典差异方法的影响。对于多尺度问题，我们发现明确方案量表的经典处理和量子处理的时间复杂性为$ \ Mathcal {o}（1/\ varepsilon）$，其中$ \ varepsilon $是缩放参数，而缩放级别的缩放级别是$ symptotic-rymptotic-resyper-proservotic-preservicotic-preserver（ap）schemes $ n of $ s $ \ y y;这表明为量子计算中的多尺度问题开发AP方案仍然非常重要。
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.
