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.
We introduce a matrix product operator (MPO) encoding of the Magnus expansion and the Dyson series for one-dimensional quantum lattice models with time-dependent Hamiltonians. The MPO construction can be made accurate up to arbitrary order in the time step, it can be applied to both finite and infinite systems, and it can handle long-range interactions. The resulting MPO can be combined with state-of-the-art time evolution algorithms based on matrix product states, allowing for drastic improvements in simulating evolution under time-dependent Hamiltonians. Our MPO construction can also be used for the optimization of quantum circuits in the context of quantum simulation of time-dependent Hamiltonians.
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.