Lindbladian Simulation with Commutator Bounds

Abstract

Trotter decomposition provides a simple approach to simulating open quantum systems by decomposing the Lindbladian into a sum of individual terms. While it is established that Trotter errors in Hamiltonian simulation depend on nested commutators of the summands, such a relationship remains poorly understood for Lindbladian dynamics. In this Letter, we derive commutator-based Trotter error bounds for Lindbladian simulation, yielding an $O(\sqrt{N})$ scaling in the number of Trotter steps for locally interacting systems on $N$ sites. When estimating observable averages, we apply Richardson extrapolation to achieve polylogarithmic precision while maintaining the commutator scaling. To bound the extrapolation remainder, we develop a general truncation bound for the Baker-Campbell-Hausdorff expansion that bypasses common convergence issues in physically relevant systems. For local Lindbladians, our results demonstrate that the Trotter-based methods outperform prior simulation techniques in system-size scaling while requiring only $O(1)$ ancillas. Numerical simulations further validate the predicted system-size and precision scaling.

Figures (4)

• Figure 1: Flowchart of our simulation framework.
• Figure 2: Trotter error for Lindbladian simulation versus system size $N$. The initial state $\rho_0={|{1^N}\rangle}{\langle{1^N}|}$. The data points are plotted on a log-log scale and fitted by linear regression. The solid lines with circle markers denote coupling strength $\gamma=0.1$, whereas the dashed lines with diamond markers denote $\gamma=1.0$. We distinguish the Trotter step $r$ by the color of the lines.
• Figure 3: Error scaling of observable expectations with and without Richardson extrapolation ($N=5$). The orange lines denote raw Trotter errors with $r = 4r_{\text{scale}}$, whereas the blue lines represent extrapolated results using $r \in \{r_{\text{scale}}, 2r_{\text{scale}}, 4r_{\text{scale}}\}$ with linear combination coefficients $\{\frac{1}{45}, \frac{-4}{9}, \frac{64}{45}\}$. The solid and dashed lines correspond to different coupling strengths $\gamma$.
• Figure 4: Trotter error for Lindbladian simulation versus system size $N$ with various initial states $\rho_0$. We fix the parameters $J=1.0$, $h=0.5$, and $t=0.2$. The data points are plotted on a log-log scale and fitted by linear regression. The solid lines with circle markers denote coupling strength $\gamma=0.1$, whereas the dashed lines with diamond markers denote $\gamma=1.0$. We distinguish the Trotter step $r$ by the color of the lines. (a) Regardless of the coupling strength $\gamma$, all the errors scale with around $O(N^{0.72})$ since the jump operator ${|{0}\rangle}_\nu{\langle{1}|}_\nu$ acts trivially on ${|{0^N}\rangle}$. (b) \ref{['fig:1']} in the main text. For $\gamma=0.1$, the errors scale with around $O(N^{0.72})$. For $\gamma=1.0$, the errors scale with around $O(N^{0.69})$. (c) The slopes are the largest and closest to commutator scaling $O(N^1)$ for the worst case. For $\gamma=0.1$, the errors scale with around $O(N^{0.79})$. For $\gamma=1.0$, the errors scale with around $O(N^{0.95})$. (d) The slopes are the smallest, and the errors decrease by an order of magnitude compared to other initial states, which might be explained by the fact that entanglement (high von Neumann entropy) accelerates quantum simulation Zhao_2025. For $\gamma=0.1$, the errors scale with around $O(N^{0.62})$. For $\gamma=1.0$, the errors scale with around $O(N^{0.61})$.

Theorems & Definitions (41)

• Theorem 1: Channel approximation
• Theorem 2: BCH truncation error bound
• Theorem 3: Observable estimation via extrapolation
• Lemma 1: see watson2025exponentially
• Lemma 2: dollard1984product
• Lemma 3
• proof
• Theorem 4
• proof
• Corollary 1