Iterative Methods for Discrete Dyson Series and Applications in Open Quantum Systems

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 deterministic iterative method for simulating system-bath dynamics. We propose two numerical schemes, which are first-order and second-order in the time step Δt, respectively. In the second-order scheme, we can safely omit 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 achieve a time complexity of O(M322KmaxKmax2) and a space complexity of O(M222KmaxKmax), where M denotes the number of system levels and Kmax 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 ≥ 3). Numerical experiments are carried out to illustrate the validity and efficiency of our method.