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arXiv:2411.03180 · Physical Review Research

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

Yu Hua Cao, Shi Jin, Nana Liu

Published Nov 19, 2025• 4 citations
arXivHTMLPDF

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.

DOI
10.1103/fkh5-b669

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We present an improved autonomization method for quantum simulation of time-dependent homogeneous dissipative linear systems, combining a clock-variable reformulation with Schrödingerization to obtain a time-independent Hamiltonian system. To control both discretization error and recovery probability, we construct the clock profile from a compactly supported window function and a normalized Dirichlet kernel: the former imposes endpoint regularity, while the latter concentrates the sampled mass and keeps the discrete normalization bounded. For the subsequent Schrödingerization step, we use an exactly periodic version of an error-function initial profile, whose explicit Fourier coefficients allow direct error estimates and simplify the analysis of smooth initialization. By combining Fourier projection with weighted recovery, we retain a single logarithmic precision factor in the full algorithm. Under finite-order derivative bounds, sampled matrix access, and exact state-preparation access, we prepare the normalized solution at time TTT to accuracy ε\varepsilonε with success probability Θ(1)Θ(1)Θ(1) using O ⁣(gαATlog⁡(gαAT/ε))\mathcal O\!\bigl(gα_AT\log(gα_AT/\varepsilon)\bigr)O(gαA​Tlog(gαA​T/ε)) matrix queries and O(g)\mathcal O(g)O(g) queries to each state-preparation oracle, where g=∥x0∥/∥x(T)∥g=\|\boldsymbol{x}_0\|/\|\boldsymbol{x}(T)\|g=∥x0​∥/∥x(T)∥ and αAα_AαA​ is the matrix-oracle normalization. We illustrate the construction and its recovery probabilities through a numerical experiment on a time-dependent two-cavity system.
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