  In this note we analyse \emph{quantitative} approximation properties of a
certain class of \emph{nonlocal} equations: Viewing the fractional heat
equation as a model problem, which involves both \emph{local} and
\emph{nonlocal} pseudodifferential operators, we study quantitative
approximation properties of solutions to it. First, relying on Runge type
arguments, we give an alternative proof of certain \emph{qualitative}
approximation results from \cite{DSV16}. Using propagation of smallness
arguments, we then provide bounds on the \emph{cost} of approximate
controllability and thus quantify the approximation properties of solutions to
the fractional heat equation. Finally, we discuss generalizations of these
results to a larger class of operators involving both local and nonlocal
contributions.
