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Unifying framework for quantum simulation algorithms for time-dependent Hamiltonian dynamics

Yu Cao1,2,3, Shi Jin1,2,3, and Nana Liu1,2,3,4,*

  • *Contact author: nana.liu@quantumlah.org
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Phys. Rev. Research 7, 043186 – Published 19 November, 2025

DOI: https://doi.org/10.1103/fkh5-b669

Abstract

Recently, there has been growing interest in simulating time-dependent Hamiltonians using quantum algorithms, driven by diverse applications, such as quantum adiabatic computing. While techniques for simulating time-independent Hamiltonian dynamics are well established, time-dependent Hamiltonian dynamics is less explored and it is unclear how to systematically organize existing methods and to find new methods. Sambe-Howland's continuous clock elegantly transforms time-dependent Hamiltonian dynamics into time-independent Hamiltonian dynamics, which means that by taking different discretizations, existing methods for time-independent Hamiltonian dynamics can be exploited for time-dependent dynamics. In this work, we systemically investigate how Sambe-Howland's clock can serve as a unifying framework for simulating time-dependent Hamiltonian dynamics. First, we demonstrate the versatility of this approach by showcasing its compatibility with analog quantum computing and digital quantum computing. Second, for digital quantum computers, we illustrate how this framework, combined with time-independent methods (e.g., product formulas, multiproduct formulas, qDrift, and linear combination of unitaries-Taylor), can facilitate the development of efficient algorithms for simulating time-dependent dynamics. This framework allows us to (1) resolve the problem of finding minimum-gate time-dependent product formulas; (2) establish a unified picture of both Suzuki's and Huyghebaert and De Raedt's approaches; (3) generalize Huyghebaert and De Raedt's first- and second-order formula to arbitrary orders; (4) answer an unsolved question in establishing time-dependent multiproduct formulas; (5) and recover continuous qDrift on the same footing as time-independent qDrift. Third, we demonstrate the efficacy of our higher-order Huyghebaert and De Raedt's algorithm through digital adiabatic simulation.

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Phys. Rev. Research 7, 043186– Published 19 November, 2025
  • Received 1 April 2025
  • Accepted 16 October 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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