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Among the single-trajectory Gaussian-based methods for solving thetime-dependent Schr\"{o}dinger equation, the variational Gaussian approximationis the most accurate one. In contrast to Heller's original thawed Gaussianapproximation, it is symplectic, conserves energy exactly, and may partiallyaccount for tunneling. However, the variational method is also much moreexpensive. To improve its efficiency, we symmetrically compose the second-ordersymplectic integrator of Faou and Lubich and obtain geometric integrators thatcan achieve an arbitrary even order of convergence in the time step. Wedemonstrate that the high-order integrators can speed up convergencedrastically compared to the second-order algorithm and, in contrast to thepopular fourth-order Runge-Kutta method, are time-reversible and conserve thenorm and the symplectic structure exactly, regardless of the time step. To showthat the method is not restricted to low-dimensional systems, we perform mostof the analysis on a non-separable twenty-dimensional model of coupled Morseoscillators. We also show that the variational method may capture tunnelingand, in general, improves accuracy over the non-variational thawed Gaussianapproximation.

High-order geometric integrators for the variational Gaussian approximation