Algebraic theory for higher-order methods in computational quantum mechanics

Abstract: We present the algebraic foundations of the symmetric Zassenhaus algorithm and some of its variants. These algorithms have proven effective in devising higher-order methods for solving the time-dependent Schr\"{o}dinger equation in the semiclassical regime. We find that the favourable properties of these methods derive directly from the structural properties of a Z2-graded Lie algebra. Commutators in this Lie algebra can be simplified explicitly, leading to commutator-free methods. Their other structural properties are crucial in proving unitarity, stability, convergence, error bounds and quadratic costs of Zassenhaus based methods. These algebraic structures have also found applications in Magnus expansion based methods for time-varying potentials where they allow significantly milder constraints for convergence and lead to highly effective schemes. The algebraic foundations laid out in this work pave the way for extending higher-order Zassenhaus and Magnus schemes to other equations of quantum mechanics.