Quantum computing has emerged as a promising avenue for achieving significant speedup, particularly in large-scale PDE simulations, compared to classical computing. One of the main quantum approaches involves utilizing Hamiltonian simulation, which is directly applicable only to Schr\"odinger-type equations. To address this limitation, Schr\"odingerisation techniques have been developed, employing the warped transformation to convert general linear PDEs into Schr\"odinger-type equations. However, despite the development of Schr\"odingerisation techniques, the explicit implementation of the corresponding quantum circuit for solving general PDEs remains to be designed. In this paper, we present detailed implementation of a quantum algorithm for general PDEs using Schr\"odingerisation techniques. We provide examples of the heat equation, and the advection equation approximated by the upwind scheme, to demonstrate the effectiveness of our approach. Complexity analysis is also carried out to demonstrate the quantum advantages of these algorithms in high dimensions over their classical counterparts.

翻译：量子计算已成为实现显著加速的重要途径，尤其在大型偏微分方程模拟方面，相较于经典计算展现出巨大潜力。主要量子方法之一涉及利用哈密顿量模拟，但该方法仅直接适用于薛定谔型方程。为突破此限制，学界发展了薛定谔化技术，通过扭曲变换将一般线性偏微分方程转化为薛定谔型方程。然而，尽管薛定谔化技术已取得进展，用于求解一般偏微分方程的相应量子电路的具体实现方案仍有待构建。本文基于薛定谔化技术，提出针对一般偏微分方程的量子算法详细实现方案。我们以热传导方程和采用迎风格式近似的平流方程为例，验证了所提方法的有效性。通过复杂度分析，论证了这些算法在高维情形下相较于经典算法的量子优势。