Farghadan Azim, Masteri Farahani Mohammad Mahdi, Akbari Mohsen
Iranian Quantum Technologies Research Center (IQTEC), Tehran, Iran.
Quantum Optics Lab, Department of Physics, Kharazmi University, Tehran, Iran.
Sci Rep. 2025 Feb 13;15(1):5317. doi: 10.1038/s41598-025-89302-8.
The numerical solution of partial differential equations (PDEs) is essential in computational physics. Over the past few decades, various quantum-based methods have been developed to formulate and solve PDEs. Solving PDEs incurs high-time complexity for high-dimensional real-world problems, and using traditional methods becomes practically inefficient. This paper presents a fast hybrid classical-quantum paradigm based on successive over-relaxation (SOR) to accelerate solving PDEs. Using the discretization method, this approach reduces the PDE solution to solving a system of linear equations, which is then addressed using the block SOR method. The block SOR method is employed to address qubit limitations, where the entire system of linear equations is decomposed into smaller subsystems. These subsystems are iteratively solved block-wise using Advantage quantum computers developed by D-Wave Systems, and the solutions are subsequently combined to obtain the overall solution. The performance of the proposed method is evaluated by solving the heat equation for a square plate with fixed boundary temperatures and comparing the results with the best existing method. Experimental results show that the proposed method can accelerate the solution of high-dimensional PDEs by using a limited number of qubits up to 2 times the existing method.
偏微分方程（PDEs）的数值解在计算物理中至关重要。在过去几十年里，人们开发了各种基于量子的方法来构建和求解偏微分方程。对于高维现实世界问题，求解偏微分方程会带来高时间复杂度，使用传统方法实际上效率很低。本文提出了一种基于逐次超松弛（SOR）的快速混合经典 - 量子范式，以加速偏微分方程的求解。通过离散化方法，该方法将偏微分方程的求解简化为求解线性方程组，然后使用块SOR方法来解决。块SOR方法用于解决量子比特限制问题，即将整个线性方程组分解为较小的子系统。这些子系统使用D-Wave Systems开发的Advantage量子计算机按块迭代求解，随后将解合并以获得整体解。通过求解具有固定边界温度的方形板的热方程，并将结果与现有最佳方法进行比较，对所提方法的性能进行了评估。实验结果表明，所提方法使用有限数量的量子比特，可将高维偏微分方程的求解速度加快至现有方法的2倍。
