  Motivated by the study of area laws for the entanglement entropy of gapped
ground states of quantum spin systems and their stability, we prove that the
unitary cocycle generated by a local time-dependent Hamiltonian can be
approximated, for any finite set $X$, by a tensor product of the corresponding
unitaries in $X$ and its complement, multiplied by a dynamics strictly
supported in the neighbourhood of the surface $\partial X$. The error decays
almost exponentially in the size of the neighbourhood and grows with the square
of the area~$\vert \partial X\vert^2$.
