  In this note we prove the exponential instability of the fractional
Calder\'on problem and thus prove the optimality of the logarithmic stability
estimate from \cite{RS17}. In order to infer this result, we follow the
strategy introduced by Mandache in \cite{M01} for the standard Calder\'on
problem. Here we exploit a close relation between the fractional Calder\'on
problem and the classical Poisson operator. Moreover, using the construction of
a suitable orthonormal basis, we also prove (almost) optimality of the Runge
approximation result for the fractional Laplacian, which was derived in
\cite{RS17}. Finally, in one dimension, we show a close relation between the
fractional Calder\'on problem and the truncated Hilbert transform.
