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有前景途径，尤其在大型偏微分方程模拟中，相较于经典计算展现出优势。主要的量子方法之一涉及利用哈密顿模拟，但该方法仅直接适用于薛定谔型方程。为解决此限制，研究者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，求解一般偏微分方程的相应量子电路的具体实现仍有待设计。本文中，我们给出了采用薛定谔化技术求解一般偏微分方程的量子算法的详细实现方案。通过热方程和采用迎风格式近似的平流方程示例，验证了所提方法的有效性。我们还进行了复杂度分析，以证明这些算法在高维问题中相较于经典算法的量子优势。