Second-order discretization of Dyson series: iterative method, numerical analysis and applications in open quantum systems

TL;DR

This work introduces discrete, second‑order Dyson‑series discretizations tailored for open quantum systems, replacing costly high‑dimensional integrals with iterative, numerically exact schemes (FRODS). The authors prove rigorous convergence rates for both first‑ and second‑order methods and implement memory‑cost reductions via bath memory truncation and coupling‑operator limitations. They demonstrate the approach on bosonic Ohmic baths and multi‑level systems (up to 50 levels), showing favorable scaling relative to established methods like i‑QuAPI and inchworm in multi‑level settings. The combination of Strang splitting interpretation, diagrammatic organization, and optimized memory handling yields a practical, scalable toolkit for simulating non‑Markovian system–bath dynamics with controlled accuracy.

Abstract

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.

Figures (10)

• Figure 1: Numbers of diagrams for various $D_{{\max}}$ and $K_{{\max}}$.
• Figure 2: Convergence test of first-order scheme on $K_{\max}$ for $\beta=5$.
• Figure 3: Convergence test of second-order scheme on $K_{\max}$ for $\beta=5$.
• Figure 4: Convergence test on $K_{\max}$ for $\beta=5$, $D_{\max}=4$, and smaller time step size $\Delta t$.
• Figure 5: Convergence test on $D_{\max}$ for $\xi=0.4,1.0$, $K_{\max}=15$, and $\Delta t = 0.1$

Theorems & Definitions (22)

• Theorem 2.1
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 4.1
• Remark 5.1
• proof
• Lemma 6.1
• proof : Proof of \ref{['lem:rho_s_extension']}
• Lemma 6.2