  A formal analysis is conducted on the exactness of various forms of unitary
coupled cluster (UCC) theory based on particle-hole excitation and
de-excitation operators. Both the conventional single exponential UCC
parameterization and a disentangled (factorized) version are considered. We
formulate a differential cluster analysis to determine the UCC amplitudes
corresponding to a general quantum state. The exactness of conventional UCC
(ability to represent any state) is explored numerically and it is formally
shown to be determined by the structure of the critical points of the UCC
exponential mapping. A family of disentangled UCC wave functions are shown to
exactly parameterize any state, thus showing how to construct
Trotter-error-free parameterizations of UCC for applications in quantum
computing. From these results, we derive an exact disentangled UCC
parameterization that employs an infinite sequence of particle-hole or general
one- and two-body substitution operators.
