  The chemistry community has long sought the exact relationship between the
conventional and the unitary coupled cluster ansatz for a single-reference
system, especially given the interest in performing quantum chemistry on
quantum computers. In this work, we show how one can use the operator
manipulations given by the exponential disentangling identity and the Hadamard
lemma to relate the factorized form of the unitary coupled-cluster
approximation to a factorized form of the conventional coupled cluster
approximation (the factorized form is required, because some amplitudes are
operator-valued and do not commute with other terms). By employing the Trotter
product formula, one can then relate the factorized form to the standard form
of the unitary coupled cluster ansatz. The operator dependence of the
factorized form of the coupled cluster approximation can also be removed at the
expense of requiring even more higher-rank operators, finally yielding the
conventional coupled cluster. The algebraic manipulations of this approach are
daunting to carry out by hand, but can be automated on a computer for small
enough systems.
