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 $p$ achieves the superconvergence of order $2p$ 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 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 $p$ achieves the superconvergence of order $2p$ 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.

arXiv:2509.22897Sep 26, 20250 citations

Quantum Algorithms for Hamiltonian Simulation

Barry C. Sanders, Graeme Ahokas, Richard Cleve +1 more

Chapman & Hall/CRC applied mathematics and nonlinear science seriesSep 19, 20070 citations

Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation

Dong An, Di Fang, Lin Lin

We propose a simple quantum algorithm for simulating highly oscillatory quantum dynamics, which does not require complicated quantum control logic for handling time-ordering operators. To our knowledge, this is the first quantum algorithm that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. Our method can be used for efficient Hamiltonian simulation in the interaction picture. In particular, we demonstrate that for the simulation of the Schrödinger equation, our method exhibits superconvergence and achieves a surprising second order convergence rate, of which the proof rests on a careful application of pseudo-differential calculus. Numerical results verify the effectiveness and the superconvergence property of our method.We propose a simple quantum algorithm for simulating highly oscillatory quantum dynamics, which does not require complicated quantum control logic for handling time-ordering operators. To our knowledge, this is the first quantum algorithm that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. Our method can be used for efficient Hamiltonian simulation in the interaction picture. In particular, we demonstrate that for the simulation of the Schrödinger equation, our method exhibits superconvergence and achieves a surprising second order convergence rate, of which the proof rests on a careful application of pseudo-differential calculus. Numerical results verify the effectiveness and the superconvergence property of our method.

arXiv:2111.03103QuantumApr 15, 202239 citations

On the efficiency of quantum algorithms for Hamiltonian simulation

Anargyros Papageorgiou, Chi Zhang

arXiv:1005.1318Quantum Information ProcessingJul 27, 201142 citations

The Floquet-Magnus expansion of unbounded operators

Daniel Burgarth, Robin Hillier, Davide Lonigro +1 more

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.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.

math-phCross-listedarXiv:2605.23734May 25, 2026

p

achieves the superconvergence of order

2p

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.