High-order commutator-free exponential time-propagation of driven quantum systems

TL;DR

This paper introduces high-order commutator-free exponential time-propagators for solving the Schrödinger equation with time-dependent Hamiltonians, offering a unitary, efficient, and scalable method suitable for large quantum systems.

Contribution

It develops and analyzes optimized high-order commutator-free propagators derived from the Magnus expansion, enhancing numerical accuracy and efficiency in quantum dynamics simulations.

Findings

Strictly preserves unitarity of quantum evolution

Enables fast and accurate simulations of large systems

Demonstrated on hydrogen atom and spin systems

Abstract

We discuss the numerical solution of the Schr\"odinger equation with a time-dependent Hamilton operator using commutator-free time-propagators. These propagators are constructed as products of exponentials of simple weighted sums of the Hamilton operator. Owing to their exponential form they strictly preserve the unitarity of time-propagation. The absence of commutators or other computationally involved operations allows for straightforward implementation and application also to large scale and sparse matrix problems. We explain the derivation of commutator-free exponential time-propagators in the context of the Magnus expansion, and provide optimized propagators up to order eight. An extensive theoretical error analysis is presented together with practical efficiency tests for different problems. Issues of practical implementation, in particular the use of the Krylov technique for the calculation of exponentials, are discussed. We demonstrate for two advanced examples, the hydrogen atom in an electric field and pumped systems of multiple interacting two-level systems or spins that this approach enables fast and accurate computations.