We propose a general strategy to discretize the Dyson series without applying direct numerical quadrature to high-dimensional integrals, and extend this framework to open quantum systems. The resulting discretization can also be interpreted as a Strang splitting combined with a Taylor expansion. Based on this formulation, we develop a numerically exact iterative method for simulation system-bath dynamics. We propose two numerical schemes, which are first-order and second-order in time step $\Delta t$ respectively. We perform a rigorous numerical analysis to establish the convergence orders of both schemes, proving that the global error decreases as $\mathcal{O}(\Delta t)$ and $\mathcal{O}(\Delta t^2)$ for the first- and second-order methods, respectively. In the second-order scheme, we can safely omitted most terms arising from the Strang splitting and Taylor expansion while maintaining second-order accuracy, leading to a substantial reduction in computational complexity. For the second-order method, we achieves a time complexity of $\mathcal{O}(M^3 2^{2K_{\max}} K_{\max}^2)$ and a space complexity of $\mathcal{O}(M^2 2^{2K_{\max}} K_{\max})$ where $M$ denotes the number of system levels and $K_{\max}$ the number of time steps within the memory length. Compared with existing methods, our approach requires substantially less memory and computational effort for multilevel systems ($M\geqslant 3$). Numerical experiments are carried out to illustrate the validity and efficiency of our method.

翻译：本文提出一种离散化戴森级数的通用策略，该方法无需对高维积分直接进行数值求积，并将此框架推广至开放量子系统。所得离散化亦可理解为Strang分裂与泰勒展开的结合。基于此表述，我们发展了一种用于模拟系统-热库动力学的数值精确迭代方法。我们提出了两种数值格式，其时间步长$\Delta t$精度分别为一阶和二阶。我们进行了严格的数值分析以确立两种格式的收敛阶，证明一阶和二阶方法的全局误差分别以$\mathcal{O}(\Delta t)$和$\mathcal{O}(\Delta t^2)$递减。在二阶格式中，我们可在保持二阶精度的前提下安全地省略由Strang分裂和泰勒展开产生的大多数项，从而显著降低计算复杂度。对于二阶方法，我们实现了$\mathcal{O}(M^3 2^{2K_{\max}} K_{\max}^2)$的时间复杂度和$\mathcal{O}(M^2 2^{2K_{\max}} K_{\max})$的空间复杂度，其中$M$表示系统能级数，$K_{\max}$表示记忆长度内的时间步数。与现有方法相比，我们的方法对于多能级系统（$M\geqslant 3$）所需的内存和计算量显著减少。数值实验验证了本方法的有效性和高效性。