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arXiv:2509.18312

A sharper Magnus expansion bound woven in binary branches

Harriet Apel, Toby S. Cubitt, Emilio Onorati

Published Sep 22, 2025• 0 citations
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Abstract

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 $O\big((A(t)/ξ)^n n^{-3/2}\big)$ on the error incurred when the Magnus expansion is truncated at an arbitrary order. The factor $n^{-3/2}$ sharpens the exponential suppression governed by the integrated norm of the generator $A(t)$ and the convergence radius $ξ$, and we prove that this dependence on the order $n$ 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.

DOI
10.48550/arxiv.2509.18312

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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.
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.
The Magnus expansion offers a powerful geometric method to express a time-ordered exponential operator solution, typical of time-dependent linear differential equations, as an ordinary operatorial exponential. This representation possesses advantageous theoretical and qualitative features, such as preserving the underlying Lie algebraic structure of the system while remaining an exact solution to the original differential equation. For any finite-dimensional Hermitian Hamiltonian governing a quantum system, the standard Magnus expansion guarantees a manifestly unitary representation at every finite order of truncation. However, this foundational property is no longer preserved if the underlying Hamiltonian is non-Hermitian. In this work, we derive a generalized, unitarized expansion framework that systematically restores and maintains the property of geometric unitarity for all bounded finite-dimensional non-Hermitian Hamiltonians.
We investigate the Magnus expansion of the NNN-operator in relativistic quantum field theory, which is related to the SSS-matrix via S=eiNS = e^{iN}S=eiN. We develop direct methods to compute matrix elements of the NNN-operator, which we refer to as Magnus amplitudes, bypassing scattering amplitudes entirely. At tree level, Magnus amplitudes are expressed in terms of retarded and advanced propagators, with each diagram weighted by factors that we identify as Murua coefficients. At loop level this structure is augmented by the Hadamard cut function, and we establish remarkable relations between loop- and tree-level Magnus amplitudes. Among these, we find that nnn-point one-loop Magnus amplitudes are entirely determined by phase-space integrals of forward limits of (n+2)(n{+}2)(n+2)-point tree-level amplitudes, and hence related to Murua coefficients, and we generalise this to a class of higher-loop contributions. Furthermore, in the case of heavy particles interacting via massless mediators, we conjecture that Magnus diagrams that contribute to the classical limit are always given by forward limits of trees, and we show this explicitly in a one-loop example. We derive these results studying theories of scalar fields with cubic interactions, but our methods are applicable to general theories as well as to integral functions appearing in gravitational-wave computations. Given that Magnus amplitudes are free of hyper-classical terms, and the known relations between Magnus amplitudes and the radial action, our results lay the groundwork for systematic and efficient calculations of classical observables from quantum field theory.
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.
Magnus expansion (ME) provides a general way to expand the real-time propagator of a time-dependent Hamiltonian within the exponential such that the unitarity is satisfied at any order. We use this property and explicit integration of Lagrange interpolation formulas for the time-dependent Hamiltonian within each time interval and derive approximations that preserve unitarity for the differential time evolution operators of general time-dependent Hamiltonians. The resulting second-order approximation is the same as using the average of Hamiltonians for two end points of time. We identify three fourth-order approximations involving commutators of Hamiltonians at different times and also derive a sixth-order expression. A test of these approximations along with other available expressions for a two-state time-dependent Hamiltonian with sinusoidal time dependences provides information on the relative performance of these approximations and suggests that the derived expressions can serve as useful numerical tools for time evolution in time-resolved spectroscopy, quantum control, quantum sensing, real-time ab initio quantum dynamics, and open system quantum dynamics.
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.