  The most efficient known quantum circuits for preparing unitary coupled
cluster states and applying Trotter steps of the arbitrary basis electronic
structure Hamiltonian involve interleaved sequences of fermionic Gaussian
circuits and Ising interaction type circuits. These circuits arise from
factorizing the two-body operators generating those unitaries as a sum of
squared one-body operators that are simulated using product formulas. We
introduce a numerical algorithm for performing this factorization that has an
iteration complexity no worse than single particle basis transformations of the
two-body operators and often results in many times fewer squared one-body
operators in the sum of squares compared to the analytical decompositions. As
an application of this numerical procedure, we demonstrate that our protocol
can be used to approximate generic unitary coupled cluster operators and
prepare the necessary high-quality initial states for techniques (like
ADAPT-VQE) that iteratively construct approximations to the ground state.
