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  • Open Access

Theory of Trotter Errors for Time-Dependent Product and Multi-product Formulas

Kaoru Mizuta1,2,3,4,*, Tatsuhiko N. Ikeda3,5,6, and Keisuke Fujii3,4,7,8

  • *Contact author: mizuta@qi.t.u-tokyo.ac.jp
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PRX Quantum 7, 033061 – Published 21 September, 2026

DOI: https://doi.org/10.1103/ywv1-7fhf

Abstract

Product formula (PF), which approximates the time evolution under a many-body Hamiltonian by the product of local time-evolution operators, is one of the central approaches for simulating quantum dynamics by quantum computers. It has been of great interest whether the errors of PFs can be expressed by commutators among local terms, since it brings the substantial suppression of the computational cost in the system size reflecting the locality of Hamiltonians. Although recent studies have revealed such error bounds for generic time-independent systems, it remains to be a difficult problem for time-dependent Hamiltonians except for low-order PFs. In this paper, we derive an error bound of generic PFs for smooth time-dependent Hamiltonians, which is expressed explicitly by nested commutators and their time derivatives. It indicates the substantial error suppression in the system size for generic local Hamiltonians, thereby giving a much better estimate of gate counts. Our derivation employs Floquet theory originally developed for time-periodic systems, with which we analyze the time-dependent PF errors by mapping them to those for time-independent PFs defined on an infinite-dimensional space. This approach is also applicable to a family of time-dependent PFs, such as the one for non-unitary dynamics. In particular, we clarify the error bound of a time-dependent multi-product formula, with which we can achieve exponentially better scaling with respect to the target accuracy than the PF. Our results will shed light on applications of near-term and future quantum computers, ranging from quantum simulation of nonequilibrium materials to faster algorithms exploiting time-dependent Hamiltonians such as adiabatic state preparation.

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Simulating dynamics of quantum many-body systems, known as Hamiltonian simulation, is one of the most promising applications of quantum computers. It is crucial for advancing physics and chemistry, such as low-temperature properties and chemical reactions, that are often impossible for classical computers to process efficiently. A central technique in Hamiltonian simulation is the product formula (PF), or “Trotterization,” which offers a simple and efficient algorithm available in near-term and future quantum computers. Our theory establishes an error bound of generic time-dependent PFs, and thereby substantially reduce the computational resource estimate of various PF-based quantum algorithms for generic time-dependent systems in a unified way. One of the major challenges is simulating time-dependent quantum systems with fewer resources. Here, we address this challenge by deriving a new error bound of the PFs for time-dependent quantum systems. Using Floquet theory developed for time-periodic Hamiltonians, our error bound achieves commutator scaling, which allows substantial error suppression in system size. It results in significant reduction in the computational costs for executing simulation with guaranteed accuracy. This approach applies to a variety of time-dependent PFs and also accelerate various PF-based quantum algorithms for time-dependent systems like the more sophisticated multi-product formula. This work offers a more cost-effective and accurate simulation of quantum systems with time-varying fields and interactions. Its application covers nonequilibrium systems like light-irradiated materials and also covers time-independent material properties via adiabatic state preparation and interaction picture. Our findings pave the way for more accessible quantum simulation by near-term and long-term quantum computers.

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PRX Quantum 7, 033061– Published 21 September, 2026
  • Received 18 November 2024
  • Revised 22 January 2026
  • Accepted 4 August 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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