  We describe an optimal procedure, as well as its efficient software
implementation, for exact and approximate synthesis of two-qubit unitary
operations into any prescribed discrete family of XX-type interactions and
local gates. This arises from the analysis and manipulation of certain
polyhedral subsets of the space of canonical gates. Using this, we analyze
which small sets of XX-type interactions cause the greatest improvement in
expected infidelity under experimentally-motivated error models. For the exact
circuit synthesis of Haar-randomly selected two-qubit operations, we find an
improvement in estimated infidelity by ~31.4% when including alongside CX its
square- and cube-roots, near to the optimal limit of ~36.9% obtained by
including all fractional applications of CX.
