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.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.

quant-pharXiv:2603.28602Mar 31, 20260 citations

Error Scaling with System Size in Digital Quantum Simulations

Natalie Pearson, Matthias Troyer, David Poulin

Bulletin of the American Physical SocietyJun 3, 20210 citations

Scaling up quantum simulations

Jürgen Berges

NatureMay 15, 20195 citations

Unbiased Hamiltonian Simulation by Reversing Trotter Error Dynamics

Keisuke Murota, Yuta Kikuchi, Enrico Rinaldi +2 more

Owing to their simplicity and low overhead, Suzuki-Trotter formulas remain the de facto Hamiltonian simulation methods on current quantum computing platforms. Systematic Trotter errors, however, will quickly become limiting when scaling to larger problems and aiming for higher accuracy. We present a mechanism that removes the systematic error of any $k$ -th order Suzuki-Trotter simulation, at the cost of a constant sampling overhead. The key insight is that the Trotter error is itself a coherent dynamics to be reversed, rather than a deviation to be bounded. By identifying the structure of this error in closed form, we carry out that reversal through quasi-probabilistic decompositions. The resulting algorithm, called Probabilistic Trotter Error Reversal (PTER), is unbiased and still improves the gate-count scaling of Suzuki-Trotter formulas while retaining their simplicity. Numerical simulations of a Heisenberg spin chain support the predicted resource advantage already at modest system sizes.Owing to their simplicity and low overhead, Suzuki-Trotter formulas remain the de facto Hamiltonian simulation methods on current quantum computing platforms. Systematic Trotter errors, however, will quickly become limiting when scaling to larger problems and aiming for higher accuracy. We present a mechanism that removes the systematic error of any $k$-th order Suzuki-Trotter simulation, at the cost of a constant sampling overhead. The key insight is that the Trotter error is itself a coherent dynamics to be reversed, rather than a deviation to be bounded. By identifying the structure of this error in closed form, we carry out that reversal through quasi-probabilistic decompositions. The resulting algorithm, called Probabilistic Trotter Error Reversal (PTER), is unbiased and still improves the gate-count scaling of Suzuki-Trotter formulas while retaining their simplicity. Numerical simulations of a Heisenberg spin chain support the predicted resource advantage already at modest system sizes.

quant-pharXiv:2606.29741Jun 30, 20260 citations

Trotter Errors and the Emergence of Chaos in Quantum Simulation

Kevin W. Kuper, Jon Pajaud, Karthik Chinni +2 more

As noisy intermediate-scale quantum (NISQ) processors increase in size and complexity, their use as general purpose quantum simulators will rely on algorithms based on the Trotter-Suzuki expansion. We run quantum simulations on a small, highly accurate quantum processor, and show how one can optimize simulation accuracy by balancing algorithmic (Trotter) errors against native errors specific to the quantum hardware at hand. We further study the interplay between native errors, Trotter errors, and the emergence of chaos as seen in measurements of a time averaged fidelity-out-of-time-ordered-correlatorAs noisy intermediate-scale quantum (NISQ) processors increase in size and complexity, their use as general purpose quantum simulators will rely on algorithms based on the Trotter-Suzuki expansion. We run quantum simulations on a small, highly accurate quantum processor, and show how one can optimize simulation accuracy by balancing algorithmic (Trotter) errors against native errors specific to the quantum hardware at hand. We further study the interplay between native errors, Trotter errors, and the emergence of chaos as seen in measurements of a time averaged fidelity-out-of-time-ordered-correlator

arXiv:2212.03843Dec 7, 20220 citations

Simple and High-Precision Hamiltonian Simulation by Compensating Trotter Error with Linear Combination of Unitary Operations

Pei Zeng, Jinzhao Sun, Liang Jiang +1 more

Trotter and linear combination of unitary (LCU) operations are two popular Hamiltonian simulation methods. The Trotter method is easy to implement and enjoys good system-size dependence endowed by commutator scaling, while the LCU method admits high-accuracy simulation with a smaller gate cost. We propose Hamiltonian simulation algorithms using LCU to compensate Trotter error, which enjoy both of their advantages. By adding few gates after the K th -order Trotter formula, we realize a better time scaling than 2 K th -order Trotter. Our first algorithm exponentially improves the accuracy scaling of the K th -order Trotter formula. For a generic Hamiltonian, the estimated gate counts of the first algorithm can be 2 orders of magnitude smaller than the best analytical bound of fourth-order Trotter formula. In the second algorithm, we consider the detailed structure of Hamiltonians and construct LCU for Trotter errors with commutator scaling. Consequently, for lattice Hamiltonians, the algorithm enjoys almost linear system-size dependence and quadratically improves the accuracy of the K th -order Trotter. For the lattice system, the second algorithm can achieve 3 to 4 orders of magnitude higher accuracy with the same gate costs as the optimal Trotter algorithm. These algorithms provide an easy-to-implement approach to achieve a low-cost and high-precision Hamiltonian simulation.Trotter and linear combination of unitary (LCU) operations are two popular Hamiltonian simulation methods. The Trotter method is easy to implement and enjoys good system-size dependence endowed by commutator scaling, while the LCU method admits high-accuracy simulation with a smaller gate cost. We propose Hamiltonian simulation algorithms using LCU to compensate Trotter error, which enjoy both of their advantages. By adding few gates after the K th -order Trotter formula, we realize a better time scaling than 2 K th -order Trotter. Our first algorithm exponentially improves the accuracy scaling of the K th -order Trotter formula. For a generic Hamiltonian, the estimated gate counts of the first algorithm can be 2 orders of magnitude smaller than the best analytical bound of fourth-order Trotter formula. In the second algorithm, we consider the detailed structure of Hamiltonians and construct LCU for Trotter errors with commutator scaling. Consequently, for lattice Hamiltonians, the algorithm enjoys almost linear system-size dependence and quadratically improves the accuracy of the K th -order Trotter. For the lattice system, the second algorithm can achieve 3 to 4 orders of magnitude higher accuracy with the same gate costs as the optimal Trotter algorithm. These algorithms provide an easy-to-implement approach to achieve a low-cost and high-precision Hamiltonian simulation.

arXiv:2212.04566PRX QuantumMar 27, 202515 citations

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.

k

-th order Suzuki-Trotter simulation, at the cost of a constant sampling overhead. The key insight is that the Trotter error is itself a coherent dynamics to be reversed, rather than a deviation to be bounded. By identifying the structure of this error in closed form, we carry out that reversal through quasi-probabilistic decompositions. The resulting algorithm, called Probabilistic Trotter Error Reversal (PTER), is unbiased and still improves the gate-count scaling of Suzuki-Trotter formulas while retaining their simplicity. Numerical simulations of a Heisenberg spin chain support the predicted resource advantage already at modest system sizes.