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Faster Algorithmic Quantum and Classical Simulations by Corrected Product Formulas

Mohsen Bagherimehrab1,2,*, Luis Mantilla Calderón2,4, Dominic W. Berry3, Philipp Schleich2,4, Mohammad Ghazi Vakili1,2, Abdulrahman Aldossary1, Jorge A. Campos Gonzalez Angulo1,4, Christoph Gorgulla5,6, and Alán Aspuru-Guzik1,2,4,7,8,9

  • *Contact author: mohsen.bagherimehrab@gmail.com
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PRX Quantum 7, 033018 – Published 28 July, 2026

DOI: https://doi.org/10.1103/phk5-slkl

Abstract

Hamiltonian simulation using product formulas is arguably the most straightforward and practical approach for algorithmic simulation of a quantum system’s dynamics on a quantum computer. Here we present corrected product formulas (CPFs), a variation of product formulas achieved by injecting auxiliary terms called correctors into standard product formulas. We establish several correctors that improve the accuracy of standard product formulas by orders of magnitude when simulating Hamiltonians comprised of two exactly simulatable partitions, a common structure of lattice Hamiltonians. Importantly, injecting these correctors increases the overall simulation cost by only a small additive or multiplicative factor. We show that correctors are particularly advantageous for perturbed systems, where one partition has a relatively small norm compared to the other, as they allow the small norm to be utilized as an additional parameter for controlling the simulation error. We demonstrate the performance of CPFs by numerical simulations for several lattice Hamiltonians. Numerical results show that our theoretical error bounds for CPFs match or outperform the empirical error scaling of standard product formulas for these systems. We also demonstrate improvements offered by CPFs by implementing small-size systems on actual quantum hardware, as well as on noisy and noiseless quantum simulators. CPFs could be a valuable algorithmic tool for early fault-tolerant quantum computers with limited computing resources. As for standard product formulas, CPFs could also be used for simulations on a classical computer.

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Popular Summary

Simulating the time evolution of quantum systems is a promising application of quantum computers. Product formulas were the first approach proposed for this task and remain a leading practical approach, despite many newer simulation techniques. They are simple, require no extra qubits or complicated operations, and often perform far better in practice. In this work, we introduce corrected product formulas by adding carefully designed correction steps to standard product formulas. These corrections substantially reduce simulation error while preserving practical simplicity, especially when part of the system acts as a small perturbation. We prove error bounds that closely match observed errors across physically relevant models and demonstrate our approach on real quantum hardware. These results show that corrected product formulas offer a practical route to accurate quantum simulations, especially on early fault-tolerant quantum computers. Beyond quantum computing, our approach can also improve classical simulations, where standard product formulas are used.

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PRX Quantum 7, 033018– Published 28 July, 2026
  • Received 28 April 2025
  • Revised 7 August 2025
  • Accepted 28 May 2026
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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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