Product formulas are a popular class of digital quantum simulation algorithms due to their conceptual simplicity, low overhead, and performance, which often exceeds theoretical expectations. Recently, Richardson extrapolation and polynomial interpolation have been proposed to mitigate the Trotter error incurred by the use of these formulas. This work provides a rigorous, general analysis of these techniques for computing time-evolved observables, simplifying the interpolation algorithm in the process, and shows that extrapolation generically improves the performance of product formulas for this task. We demonstrate that, to achieve error ϵ in a simulation of time T using a p th -order product formula with extrapolation, circuit depths of O ( T 1 + 1 / p polylog ( 1 / ϵ ) ) are sufficient—an exponential improvement in the precision over product formulas alone. Furthermore, we prove that these algorithms achieve commutator scaling, and improve the T complexity for the interpolation algorithm. By relaxing the requirement of performing exact Chebyshev interpolation, our simplified algorithm eliminates the need for fractional implementations of Trotter steps, reducing computational overhead. Finally, we show these techniques can be combined with the classical shadows method to estimate many time-evolved local observables. Taken together, our findings provide the strongest evidence yet for the utility of Trotter error-mitigation techniques in algorithmic applications.Product formulas are a popular class of digital quantum simulation algorithms due to their conceptual simplicity, low overhead, and performance, which often exceeds theoretical expectations. Recently, Richardson extrapolation and polynomial interpolation have been proposed to mitigate the Trotter error incurred by the use of these formulas. This work provides a rigorous, general analysis of these techniques for computing time-evolved observables, simplifying the interpolation algorithm in the process, and shows that extrapolation generically improves the performance of product formulas for this task. We demonstrate that, to achieve error ϵ in a simulation of time T using a p th -order product formula with extrapolation, circuit depths of O ( T 1 + 1 / p polylog ( 1 / ϵ ) ) are sufficient—an exponential improvement in the precision over product formulas alone. Furthermore, we prove that these algorithms achieve commutator scaling, and improve the T complexity for the interpolation algorithm. By relaxing the requirement of performing exact Chebyshev interpolation, our simplified algorithm eliminates the need for fractional implementations of Trotter steps, reducing computational overhead. Finally, we show these techniques can be combined with the classical shadows method to estimate many time-evolved local observables. Taken together, our findings provide the strongest evidence yet for the utility of Trotter error-mitigation techniques in algorithmic applications.

arXiv:2408.14385PRX QuantumJul 2, 202512 citations

CALCULUS OF THE TROTTER APPROXIMATION

Giuseppe Da Prato, Luciano Tubaro

CRC Press eBooksAug 19, 19940 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

Mathematical Interlude

Wim Magnus, Wim Schoenmaker

Springer series in solid-state sciencesJan 1, 20020 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

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.