Second-order discretization of Dyson series: iterative method, numerical analysis and applications in open quantum systems
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912654069596160 |
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| author | Cai, Zhenning Sun, Yixiao Wang, Geshuo |
| author_facet | Cai, Zhenning Sun, Yixiao Wang, Geshuo |
| contents | 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 $Δt$ respectively. We perform a rigorous numerical analysis to establish the convergence orders of both schemes, proving that the global error decreases as $\mathcal{O}(Δt)$ and $\mathcal{O}(Δt^2)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15287 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Second-order discretization of Dyson series: iterative method, numerical analysis and applications in open quantum systems Cai, Zhenning Sun, Yixiao Wang, Geshuo Quantum Physics Numerical Analysis 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 $Δt$ respectively. We perform a rigorous numerical analysis to establish the convergence orders of both schemes, proving that the global error decreases as $\mathcal{O}(Δt)$ and $\mathcal{O}(Δt^2)$ 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. |
| title | Second-order discretization of Dyson series: iterative method, numerical analysis and applications in open quantum systems |
| topic | Quantum Physics Numerical Analysis |
| url | https://arxiv.org/abs/2510.15287 |