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.

中文翻译：

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将 Lindblad 方程转换为实值线性方程组：算法的性能优化和并行化


随着峰值性能和内存容量的不断增加，现代超级计算机为开放多体量子系统的数值研究提供了新的视角。这些系统通常使用描述系统密度算子演化的马尔可夫量子主方程来建模。在本文中，我们讨论 Lindblad 形式的主方程，这是量子光学、腔量子电动力学和光力学中流行的理论工具。通过使用广义盖尔曼矩阵作为基础，任何 Lindblad 方程都可以转化为具有实数系数的常微分方程组。最近，我们提出了一种计算复杂度为 O(N5logN) 的变换实现，对于密集 Lindbaldians 缩放为 O(N5logN)，对于稀疏 Lindbaldians 缩放为 O(N3logN)。然而，不可行的内存成本仍然是大型模型发展的一个严重障碍。在这里，我们提出了一种基于并行集群的算法实现，并证明它允许我们使用 25 个节点和 64 GB 来集成维度 N=2000 的稀疏 Lindbladian 模型和维度 N=200 的密集随机 Lindbladian 模型每个节点的 RAM。