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.
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