  The unitary coupled cluster (UCC) approximation is one of the more promising
wave-function ans\"atze for electronic structure calculations on quantum
computers via the variational quantum eigensolver algorithm. However, for large
systems with many orbitals, the required number of UCC factors still leads to
very deep quantum circuits, which can be challenging to implement. Based on the
observation that most UCC amplitudes are small for weakly correlated molecules,
we devise an algorithm that employs a Taylor expansion in the small amplitudes,
trading off circuit depth for extra measurements. Strong correlations can be
taken into account by performing the expansion about a small set of UCC
factors, which are treated exactly. Near equilibrium, the Taylor series
expansion often works well without the need to include any exact factors; as
the molecule is stretched and correlations increase, we find only a small
number of factors need to be treated exactly.
