Meyerov Iosif, Kozinov Evgeny, Liniov Alexey, Volokitin Valentin, Yusipov Igor, Ivanchenko Mikhail, Denisov Sergey
Mathematical Center, Lobachevsky University, 603950 Nizhni Novgorod, Russia.
Department of Applied Mathematics, Lobachevsky University, 603950 Nizhni Novgorod, Russia.
Entropy (Basel). 2020 Oct 6;22(10):1133. doi: 10.3390/e22101133.
With their constantly increasing peak performance and memory capacity, modern supercomputers offer new perspectives on numerical studies of open many-body quantum systems. These systems are often modeled by using Markovian quantum master equations describing the evolution of the system density operators. In this paper, we address master equations of the Lindblad form, which are a popular theoretical tools in quantum optics, cavity quantum electrodynamics, and optomechanics. By using the generalized Gell-Mann matrices as a basis, any Lindblad equation can be transformed into a system of ordinary differential equations with real coefficients. Recently, we presented an implementation of the transformation with the computational complexity, scaling as O(N5logN) for dense Lindbaldians and O(N3logN) for sparse ones. However, infeasible memory costs remains a serious obstacle on the way to large models. Here, we present a parallel cluster-based implementation of the algorithm and demonstrate that it allows us to integrate a sparse Lindbladian model of the dimension N=2000 and a dense random Lindbladian model of the dimension N=200 by using 25 nodes with 64 GB RAM per node.
随着现代超级计算机的峰值性能和内存容量不断提高，它们为开放多体量子系统的数值研究提供了新的视角。这些系统通常通过使用描述系统密度算符演化的马尔可夫量子主方程来建模。在本文中，我们讨论林德布拉德形式的主方程，它是量子光学、腔量子电动力学和光机械学中常用的理论工具。通过使用广义盖尔曼矩阵作为基，任何林德布拉德方程都可以转化为具有实系数的常微分方程组。最近，我们给出了一种变换的实现，对于密集林德布拉德算符，其计算复杂度按O(N5logN) 缩放，对于稀疏林德布拉德算符，按O(N3logN) 缩放。然而，内存成本过高仍然是构建大型模型的严重障碍。在这里，我们给出了基于并行集群的算法实现，并证明通过使用25个节点（每个节点有64GB内存），我们能够对维度为N = 2000的稀疏林德布拉德模型和维度为N = 200的密集随机林德布拉德模型进行积分。
