  We explicitly compute the optimal cost for a class of example problems in
geometric quantum control. These problems are defined by a Cartan decomposition
of $su(2^n)$ into orthogonal subspaces $\mathfrak{l}$ and $\mathfrak{p}$ such
that $[\mathfrak{l},\mathfrak{l}] \subseteq \mathfrak{p},
[\mathfrak{p},\mathfrak{l}] = \mathfrak{p}, [\mathfrak{p},\mathfrak{p}]
\subseteq \mathfrak{l}$. Motion in the $\mathfrak{l}$ direction are assumed to
have negligible cost, where motion in the $\mathfrak{p}$ direction do not. In
the special case of two qubits, our results correspond to the minimal
interaction cost of a given unitary.
