Reducing the Gate Count with Efficient Trotter-Suzuki Schemes

TL;DR

This work outlines a short guide to Trotter-Suzuki schemes and their implementations in general, and highlights new efficient schemes found by the optimization framework, and demonstrates their performance on the Heisenberg model.

Abstract

Hamiltonian formulations of lattice field theories provide access to real-time dynamics, but their simulation is difficult to implement efficiently. Trotter-Suzuki decompositions are at the center of time evolution computation, either on quantum hardware or classically, for instance with the use of tensor networks. While low-order Trotterizations remain the standard choice due to their simplicity, higher-order schemes offer the potential for improved efficiency. In this work we outline a short guide to Trotter-Suzuki schemes and their implementations in general. To help with this, we highlight new efficient schemes found by our optimization framework, and demonstrate their performance on the Heisenberg model.

Figures (3)

• 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 either side, 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). 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, and 6 minima in total. 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.
• Figure 3: Presented are the errors of Trotterized time evolution in the Heisenberg XXZ model approximated by the Frobenius norm $\Delta_{n}^{\textrm{exp}}$ (see Eq. \ref{['eq:Frobenius']}), for a collection of historical Trotterizations and our two recommended schemes at orders $n = 4, 6$ (see Tab. \ref{['tab:recommened6']} and \ref{['tab:recommened14']}). On the left-hand side we observe the error as a function of the system size $L$, and find that it plateaus towards the thermodynamic limit. We present the improved efficiency of our novel schemes with respect to the computational cost $q N_{t}$, and find that our $n = 6$ scheme performs better than the historical schemes in a large region of the cost. The data was simulated at total time $t = 10$, and the gray lines indicate where the two plots were simulated at, i.e. $q N_{t} = 500$ and $L = 6$.