  For a native gate set which includes all single-qubit gates, we apply results
from symplectic geometry to analyze the spaces of two-qubit programs accessible
within a fixed number of gates. These techniques yield an explicit description
of this subspace as a convex polytope, presented by a family of linear
inequalities themselves accessible via a finite calculation. We completely
describe this family of inequalities in a variety of familiar example cases,
and as a consequence we highlight a certain member of the "XY-family" for which
this subspace is particularly large, i.e., for which many two-qubit programs
admit expression as low-depth circuits.
