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"). Our detailed analyses show that for the heat and linear hyperbolic equations the quantum difference methods provide exponential speedup over the classical difference method with respect to the spatial dimension. For the multiscale problem, the time complexity of both the classical treatment and quantum treatment for the explicit scheme scales as $O(1/\varepsilon)$, where $\varepsilon$ is the scaling parameter, while the scaling for the 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：我们详细分析表明,对于热量和线性双曲方程式,量差法的量差法比传统的空间维度差异法的指数加速。对于多尺度问题,典型治疗和量子处理方法的时间复杂度分别为O(1/\varepsilon)美元($@varepsilon)美元($@varepsilon)和量子处理,其中1美元是缩放参数),而Asymptic-Preservation(AP)计划的规模不取决于$\varepsilon($)。这表明,为量子计算中的多尺度问题制定AP计划仍然非常重要。