Efficient Trotter-Suzuki Schemes for Long-time Quantum Dynamics

TL;DR

This work tackles the problem of accurately simulating long-time quantum dynamics by developing a framework to construct high-order Trotter-Suzuki decompositions directly from scratch, optimizing their parameters to minimize leading-time errors. Central to the approach is a recursive, symmetric BCH-based construction that yields error coefficients (e.g., $O_3=\alpha C_1+\beta C_2$ with $C_1=[A,[A,B]]$, $C_2=[B,[B,A]]$) and an objective function Err$_n$ whose minimization—via Levenberg–Marquardt on parameters $a_i,b_i$—produces highly efficient schemes for given cycle counts $q$. The authors demonstrate two novel schemes at $n=4$ with $q=6$ and at $n=6$ with $q=14$, showing improved theoretical efficiency and practical performance on the Heisenberg XXZ model and the quantum harmonic oscillator; they also reveal that schemes with coefficients more uniform and closer to an origin point tend to accumulate error more gently over long times. A supplementary practical framework improvement explores adding a penalty for deviation from the origin, improving correlation with observed errors in some cases, though the benefits are model-dependent. The work provides a software and data pipeline for reproducing the optimizations and results, and offers directions toward higher-order schemes, complex nonunitary schemes, and hardware-oriented implementations.

Abstract

Accurately simulating long-time dynamics of many-body systems is a challenge in both classical and quantum computing due to the accumulation of Trotter errors. While low-order Trotter-Suzuki decompositions are straightforward to implement, their rapidly growing error limits access to long-time observables. We present a framework for constructing efficient high-order Trotter-Suzuki schemes by identifying their structure and directly optimizing their parameters over a high-dimensional space. This method enables the discovery of new schemes with significantly improved efficiency compared to traditional constructions, such as those by Suzuki and Yoshida. Based on the theoretical efficiency and practical performance, we recommend two novel highly efficient schemes at $4^{\textrm{th}}$ and $6^{\textrm{th}}$ order. We also demonstrate the effectiveness of these decompositions on the Heisenberg model and the quantum harmonic oscillator, and find that for a fixed final time they perform better across the computational cost. Even when using large time steps, they surpass established low-order schemes like the Leapfrog. Finally, we investigate the in-practice performance of different Trotter schemes and find the decompositions with more uniform coefficients tend to feature improved error accumulation over long times. We have included this observation into our choice of recommended schemes.

Figures (11)

• Figure 1: Representation of a Trotter-Suzuki scheme with an arbitrary number of stages $\Lambda$. There are $q$ cycles, each consisting of two ramps. Ramps forward are indicated by the purple line, while pink lines represent the backward ramps. Read from left to right, while multiplying exponents of operators $A_{k}$ with appropriate parameters $c_{i}$ or $d_{i}$, one obtains the decomposition from Eq. \ref{['eq:general_scheme']}. Adapted from Ostmeyer:2022.
• Figure 2: Error manifolds of $2^{\textrm{nd}}$ order schemes at $q = 2$ cycles (left) and $4^{\textrm{th}}$ order schemes at $q = 4$ cycles (right). In both cases there is one real free parameter. The error function for 2 cycles is a simple one, with a single minimum, which is not hard to minimize. We plot it as a star, as well as the Leapfrog scheme, which can be found at null free parameter. The picture is more complicated at $q = 4$ cycles, where one finds 3 branches, two of them merging into a complex-valued parameter region. This manifold has 6 minima in total, which is already harder to optimize than at order $n = 2$. We plot the global minimum again, and present the scheme by Forest & Ruth on the real branch, where the value of the free parameter reaches zero. More manifold visualizations at $q = 5$ and $q = 6$ cycles are available in our repository markomalezic_2026_18347430. We obtained the plots by solving the constraints and computing the error functions with respect to the free parameter.
• Figure 3: Theoretical efficiency $\textrm{Eff}_{n}$ according to Eq. \ref{['eq:eff']} at orders $n = 2, 4, 6$ across the relevant number of cycles $q$ for a collection of historical schemes and our novel decompositions. We find a plateau towards the maximal number of cycles in a given order. For our novel schemes we plot a recommended one according to its theoretical efficiency and consistent performance in practice. In order to compare our best schemes, we also plot the scheme with the highest $\textrm{Eff}_{n}$, unless that is our recommended one in which case we plot the theoretically next best decomposition. From the results at the maximal no. cycles we find a slight theoretical improvement of our schemes in both orders $n = 4$ and $n = 6$ over those by Blanes & Moan BLANES2002313. We note that the efficiency values are not comparable between orders.
• Figure 4: Experimental efficiency $\textrm{Eff}_{n}^{\textrm{exp}}$\ref{['eq:exp_eff']} of the Heisenberg XXZ model with $L = 6$ spins at orders $n = 2, 4, 6$ across the relevant number of cycles $q$ for a collection of historical schemes and our novel schemes. We also investigated two fundamentally different scheme orderings -- local (top) and global (bottom), and found a similar picture to that of the theoretical efficiency (see Fig. \ref{['fig:theo_efficiency']}). For our novel schemes we plot a recommended one according to its theoretical efficiency and consistent performance in practice, along with the best scheme. In the global grouping we find a few schemes which turn out to perform better in some cases, but are expected to perform inconsistently in other models. The data was simulated for total evolution time $t = 10$ at a fixed computational cost $q N_{t} = 1000$. The experimental efficiency is rescaled, such that it mimics the theoretical efficiency. We note that the efficiency values are not comparable between orders and models.
• Figure 5: Experimental error $\Delta_{n}^{\textrm{exp}}$ from numerical simulations of the Heisenberg XXZ model with $L = 6$ spins in the local grouping $S^{(3 L)}$ (left) and the global grouping $S^{(3)}$ (right) as a function of the computational cost $q N_{t}$. The data was simulated at a fixed evolution time $t = 10$. We plot a collection of historically relevant schemes and compare them to our recommended order $n = 4$ and $n = 6$ schemes. On the lower end of the cost we find a plateau, which appears due to the properties of the Frobenius norm that defines the accumulated error. The relevant information is in the scaling region that comes after the plateau, which increases in steepness with higher orders $n$. Here we find that our recommended order $n = 6$ scheme at $q = 14$ cycles performs better than the rest of the known schemes at each cost for orders $n = 2, 4, 6$ and a significant part of the cost for the best known order $n = 8, 10$ schemes by Morales et al. morales2022greatly.