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：量子计算已成为相对于经典计算实现显著加速的潜在途径，尤其是在大规模偏微分方程模拟中。主要的量子方法之一是利用哈密顿量模拟技术，该技术仅直接适用于薛定谔型方程。为克服这一局限，研究者开发了薛定谔化方法，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，求解一般偏微分方程的相应量子电路尚未完成显式设计。本文提出利用薛定谔化技术求解一般偏微分方程的量子算法详细实现方案。我们以热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。同时进行的复杂度分析表明，这些算法在高维场景下相较于经典算法具有量子优越性。