  The explicit split-operator algorithm has been extensively used for solving
not only linear but also nonlinear time-dependent Schr\"{o}dinger equations.
When applied to the nonlinear Gross-Pitaevskii equation, the method remains
time-reversible, norm-conserving, and retains its second-order accuracy in the
time step. However, this algorithm is not suitable for all types of nonlinear
Schr\"{o}dinger equations. Indeed, we demonstrate that local control theory, a
technique for the quantum control of a molecular state, translates into a
nonlinear Schr\"{o}dinger equation with a more general nonlinearity, for which
the explicit split-operator algorithm loses time reversibility and efficiency
(because it has only first-order accuracy). Similarly, the trapezoidal rule
(the Crank-Nicolson method), while time-reversible, does not conserve the norm
of the state propagated by a nonlinear Schr\"{o}dinger equation. To overcome
these issues, we present high-order geometric integrators suitable for general
time-dependent nonlinear Schr\"{o}dinger equations and also applicable to
nonseparable Hamiltonians. These integrators, based on the symmetric
compositions of the implicit midpoint method, are both norm-conserving and
time-reversible. The geometric properties of the integrators are proven
analytically and demonstrated numerically on the local control of a
two-dimensional model of retinal. For highly accurate calculations, the
higher-order integrators are more efficient. For example, for a wavefunction
error of $10^{-9}$, using the eighth-order algorithm yields a $48$-fold speedup
over the second-order implicit midpoint method and trapezoidal rule, and
$400000$-fold speedup over the explicit split-operator algorithm.

Chemical Physics;Computational Physics;Quantum Physics