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  • Open Access

Exponentially Reduced Circuit Depths Using Trotter Error Mitigation

James D. Watson1,2 and Jacob Watkins3,4,*

  • *Contact author: jacob.a.watkins@jpmchase.com
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PRX Quantum 6, 030325 – Published 12 August, 2025

DOI: https://doi.org/10.1103/kw39-yxq5

Abstract

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 pth-order product formula with extrapolation, circuit depths of O(T1+1/ppolylog(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.

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Popular Summary

Quantum computers can simulate the dynamics of complex quantum-mechanical systems at speeds much faster than traditional “classical” computers. However, classical computers can still be used alongside quantum computers to improve the quality of these calculations. For example, there are several methods for improving the accuracy of quantum computing simulations, which involve combining the results of calculations for several different values of an accuracy parameter. This work unifies these approaches and provides a thorough mathematical analysis of their performance, showing that they perform similarly well to one another and are all effective at reducing the quantum computing resources needed to achieve a desired quality of result.

In particular, we consider Trotter-based methods for quantum simulation, a popular set of techniques due to their simplicity and high performance. Accuracy is determined by the number of iterations, and we vary this number to collect data to be used for classical data analysis. We prove that maximum run times can be exponentially reduced using one of two considered extrapolation techniques, Richardson or polynomial, compared to using Trotter alone. Moreover, these extrapolation techniques have the beneficial property that the error tends to zero as the Hamiltonian operator terms tend toward commuting.

Beyond the two techniques considered here, our work indicates the value of more general schemes that learn the behavior of Trotter simulations with respect to accuracy parameters. Concrete demonstrations of the protocol on actual quantum hardware will help assess the real value of these tools for simulating quantum systems studied in scientific and industrial contexts.

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PRX Quantum 6, 030325– Published 12 August, 2025
  • Received 24 October 2024
  • Revised 21 May 2025
  • Accepted 2 July 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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