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arXiv:2509.06054 · npj Quantum Information

High-order Magnus expansion for Hamiltonian simulation

Di Fang, Diyi Liu, Shuchen Zhu

Published Oct 7, 2026• 0 citations
arXivHTMLPDF

Abstract

Efficient simulation of quantum dynamics with time-dependent Hamiltonians is important not only for time-varying systems but also for time-independent Hamiltonians in the interaction picture. Such simulations are more challenging than their time-independent counterparts due to the complexity introduced by time ordering. Existing algorithms that aim to capture commutator-based scaling either exhibit polynomial cost dependence on the Hamiltonian’s time derivatives or are limited to low-order accuracy. In this work, we establish the general commutator-scaling error bounds for the truncated Magnus expansion at arbitrary order, where only Hamiltonian terms appear in the nested commutators, with no time derivatives involved. Building on this analysis, we design a high-order quantum algorithm with explicit circuit constructions. The algorithm achieves cost scaling with the commutator structure in the high-precision regime and depends only logarithmically on the Hamiltonian’s time variation, making it efficient under the standard input model for general time-dependent settings, including Hamiltonians arising in the interaction picture.

DOI
10.1038/s41534-026-01385-x

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Efficient simulation of quantum dynamics with time-dependent Hamiltonians is important not only for time-varying systems but also for time-independent Hamiltonians in the interaction picture. Such simulations are more challenging than their time-independent counterparts due to the complexity introduced by time ordering. Existing algorithms that aim to capture commutator-based scaling either exhibit polynomial cost dependence on the Hamiltonian’s time derivatives or are limited to low-order accuracy. In this work, we establish the general commutator-scaling error bounds for the truncated Magnus expansion at arbitrary order, where only Hamiltonian terms appear in the nested commutators, with no time derivatives involved. Building on this analysis, we design a high-order quantum algorithm with explicit circuit constructions. The algorithm achieves cost scaling with the commutator structure in the high-precision regime and depends only logarithmically on the Hamiltonian’s time variation, making it efficient under the standard input model for general time-dependent settings, including Hamiltonians arising in the interaction picture.
We develop a fourth-order Magnus expansion based quantum algorithm for the simulation of many-body problems involving two-level quantum systems with time-dependent Hamiltonians, H(t)\mathcal{H}(t)H(t). A major hurdle in the utilization of the Magnus expansion is the appearance of a commutator term which leads to prohibitively long circuits. We present a technique for eliminating this commutator and find that a single time-step of the resulting algorithm is only marginally costlier than that required for time-stepping with a time-independent Hamiltonian, requiring only three additional single-qubit layers. For a large class of Hamiltonians appearing in liquid-state nuclear magnetic resonance (NMR) applications, we further exploit symmetries of the Hamiltonian and achieve a surprising reduction in the expansion, whereby a single time-step of our fourth-order method has a circuit structure and cost that is identical to that required for a fourth-order Trotterized time-stepping procedure for time-independent Hamiltonians. Moreover, our algorithms are able to take time-steps that are larger than the wavelength of oscillation of the time-dependent Hamiltonian, making them particularly suited for highly-oscillatory controls. The resulting quantum circuits have shorter depths for all levels of accuracy when compared to first and second-order Trotterized methods, as well as other fourth-order Trotterized methods, making the proposed algorithm a suitable candidate for simulation of time-dependent Hamiltonians on near-term quantum devices.
The Magnus expansion provides an exponential representation of one-parameter operator families, expressed as a series expansion in their generators. This is particularly relevant in quantum mechanics for describing a unitary evolution determined by a time-dependent Hamiltonian generator of the dynamics. The solution is constructed as a series expansion in terms of increasingly complex nested commutators that rapidly become challenging to compute directly. This work establishes a universal upper bound scaling as O((A(t)/ξ)nn−3/2)O\big((A(t)/ξ)^n n^{-3/2}\big)O((A(t)/ξ)nn−3/2) on the error incurred when the Magnus expansion is truncated at an arbitrary order. The factor n−3/2n^{-3/2}n−3/2 sharpens the exponential suppression governed by the integrated norm of the generator A(t)A(t)A(t) and the convergence radius ξξξ, and we prove that this dependence on the order nnn is optimal for the corresponding majorant. The main technical ingredient of the proof is the binary tree representation introduced by Iserles and Norsett from which we derive a recursion formula to delimit the magnitude of any term in the expansion. We complement our analytic bound with explicit computation of the first tree coefficients. With these findings we aim to contribute to the understanding of the accuracy and limitations of the Magnus expansion technique, and to provide a sharper bound for approximating quantum dynamics when information on the generator is not available.
The evolution of a quantum system under time-dependent driving exhibits phenomena that are absent in its stationary counterpart. However, the high dimensionality and non-commutative nature of quantum dynamics make this a challenging problem. The Magnus expansion provides an analytic framework to approximate the effective dynamics on short time-scales, but computing high-order terms with existing methods is computationally expensive. We introduce a scalable approach that reduces the computational effort to depend only on the degrees of freedom defining the time-dependent control function. We focus specifically on Hamiltonians consisting of a constant drift term and a controllable term. Our method provides a polynomial expression for the Magnus expansion which can be evaluated several orders of magnitude faster than previous techniques, enabling broad applications in the realms of quantum simulation and quantum optimal control. We showcase an application of the method by designing control pulses for the 5-qubit phase gate on a neutral-atom platform utilizing Rydberg atoms.
The Magnus expansion has long been a celebrated subject in numerical analysis, leading to the development of many useful classical integrators. More recently, it has been discovered to be a powerful tool for designing quantum algorithms for Hamiltonian simulation in quantum computing. In particular, surprising superconvergence behavior has been observed for quantum Magnus algorithms applied to the simulation of the Schrödinger equation, with the first- and second-order methods exhibiting doubled convergence order. In this work, we provide a rigorous proof that such superconvergence extends to general high-order quantum Magnus algorithms. Specifically, we show that a quantum Magnus algorithm of order ppp achieves the superconvergence of order 2p2p2p in time when applying to the Schrödinger equation simulation in the interaction picture. Our analysis combines techniques from semiclassical analysis and Weyl calculus, offering a new perspective on the mathematical foundations of quantum algorithms for time-dependent Hamiltonian simulation.
The Floquet-Magnus expansion is a widely used tool to derive effective descriptions of time-periodic quantum systems by approximating their dynamics with a time-independent Hamiltonian. However, its standard formulation is, strictly speaking, restricted to bounded Hamiltonians. In this work, we extend its definition and analysis to a broad class of time-periodic unbounded Hamiltonians. Our approach is based on an a priori distinct nonperturbative framework for the construction of effective Hamiltonians, which we show to reproduce the Floquet-Magnus expansion. A particular strength of our framework is that it allows us to prove that the resulting effective dynamics approximates the original time evolution propagators to arbitrary order in the high-frequency limit without requiring convergence of the Floquet-Magnus expansion, a condition that is already highly restrictive even in the bounded setting. We illustrate the scope of the method on representative models: the quantum Rabi Hamiltonian in the interaction picture, and the periodically driven quantum harmonic oscillator.
Hamiltonian simulation is arguably the most fundamental application of quantum computers. The Magnus operator is a popular method for time-dependent Hamiltonian simulation in computational mathematics, yet its usage requires the implementation of exponentials of commutators, which has previously made it unappealing for quantum computing. The development of commutator-free quasi-Magnus operators (CFQMs) circumvents this obstacle, at the expense of a lack of provable global numeric error bounds. In this work, we establish one such error bound for CFQM-based time-dependent quantum Hamiltonian simulation by carefully estimating the error of each step involved in their definition. This allows us to compare its cost with the alternatives, and show that CFQMs are often the most efficient product-formula technique available by more than an order of magnitude. As a result, we find that CFQMs may be particularly useful to simulate time-dependent Hamiltonians on early fault-tolerant quantum computers.