  We consider the dynamics $t\mapsto\tau_t$ of an infinite quantum lattice
system that is generated by a local interaction. If the interaction decomposes
into a finite number of terms that are themselves local interactions, we show
that $\tau_t$ can be efficiently approximated by a product of $n$
automorphisms, each of them being an alternating product generated by the
individual terms. For any integer $m$, we construct a product formula (in the
spirit of Trotter) such that the approximation error scales as $n^{-m}$. Our
bounds hold in norm, pointwise for algebra elements that are sufficiently well
approximated by finite volume observables.
