论文标题
水文线性系统的量子计算和预处理
Quantum Computing and Preconditioners for Hydrological Linear Systems
论文作者
论文摘要
建模水文断裂网络是计算地球科学中的标志性挑战。准确预测断裂系统的关键特征，例如渗透需要求解远远超出当前或将来的高性能功能的大型线性系统。从理论上讲，量子计算机可以绕过经典方法面临的内存和速度约束，但是首先必须解决一些技术问题。这些困难中的主要是，这种系统通常是条件不足的，即系统的小变化可以在溶液中产生巨大的变化，这可以减慢线性求解算法的性能。我们测试了几种现有的量子技术以改善状态数量，但发现它们不足。然后，我们介绍了逆拉普拉斯预处理程序，该逆向前提器将系统的条件数从$ O（n）$提高到$ o（\ sqrt {n}）$，并承认量子实现。这些结果是为断裂系统开发量子求解器的关键第一步，既可以推进水文建模的状态，又为量子线性系统算法提供了新颖的现实世界应用。
Modeling hydrological fracture networks is a hallmark challenge in computational earth sciences. Accurately predicting critical features of fracture systems, e.g. percolation, can require solving large linear systems far beyond current or future high performance capabilities. Quantum computers can theoretically bypass the memory and speed constraints faced by classical approaches, however several technical issues must first be addressed. Chief amongst these difficulties is that such systems are often ill-conditioned, i.e. small changes in the system can produce large changes in the solution, which can slow down the performance of linear solving algorithms. We test several existing quantum techniques to improve the condition number, but find they are insufficient. We then introduce the inverse Laplacian preconditioner, which improves the scaling of the condition number of the system from $O(N)$ to $O(\sqrt{N})$ and admits a quantum implementation. These results are a critical first step in developing a quantum solver for fracture systems, both advancing the state of hydrological modeling and providing a novel real-world application for quantum linear systems algorithms.
