  In this note we study the properties of a sequence of approximate propagators
for the Schr\"odinger equation, in the spirit of Feynman's path integrals.
Precisely, we consider Hamiltonian operators arising as the Weyl quantization
of a quadratic form in phase space, plus a bounded potential perturbation in
the form of a pseudodifferential operator with a rough symbol. It is known that
the corresponding Schr\"odinger propagator is a generalized metaplectic
operator. This naturally motivates the introduction of a manageable time
slicing approximation consisting of operators of the same type. By means of
techniques and function spaces of time-frequency analysis it is possible to
obtain several convergence results with precise rates in terms of the mesh size
of the time slicing subdivision. In particular, we prove convergence in the
norm operator topology in $L^2$, as well as pointwise convergence of the
corresponding integral kernels for non-exceptional times.
