Superconvergence of High-order Magnus Quantum Algorithms

Abstract: 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 Schr\"odinger 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 Schr\"odinger 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.