  A quantum channel from a system $A$ of dimension $d_A$ to a system $B$ of
dimension $d_B$ is a completely positive trace-preserving map from complex
$d_A\times d_A$ to $d_B\times d_B$ matrices, and the set of all such maps with
Kraus rank $r$ has the structure of a smooth manifold. We describe this set in
two ways. First, as a quotient space of (a subset of) the $rd_B\times d_A$
dimensional Stiefel manifold. Secondly, as the set of all Choi-states of a
fixed rank $r$. These two descriptions are topologically equivalent. This
allows us to show that the set of all Choi-states corresponding to extreme
channels from system $A$ to system $B$ of a fixed Kraus rank $r$ is a smooth
submanifold of dimension $2rd_Ad_B-d_A^2-r^2$ of the set of all Choi-states of
rank $r$. As an application, we derive a lower bound on the number of
parameters required for a quantum circuit topology to be able to approximate
all extreme channels from $A$ to $B$ arbitrarily well.
