  The variational quantum eigensolver is one of the most promising algorithms
for near-term quantum computers. It has the potential to solve quantum
chemistry problems involving strongly correlated electrons, which are otherwise
difficult to solve on classical computers. The variational eigenstate is
constructed from a number of factorized unitary coupled-cluster terms applied
onto an initial (single-reference) state. Current algorithms for applying one
of these operators to a quantum state require a number of operations that
scales exponentially with the rank of the operator. We exploit a hidden SU($2$)
symmetry to allow us to employ the linear combination of unitaries approach,
Our \textsc{Prepare} subroutine uses $n+2$ ancilla qubits for a rank-$n$
operator. Our \textsc{Select}($\hat U$) scheme uses $\mathcal{O}(n)$
\textsc{Cnot} gates. This results in an full algorithm that scales like the
cube of the rank of the operator $n^3$, a significant reduction in complexity
for rank five or higher operators. This approach, when combined with other
algorithms for lower-rank operators (when compared to the standard
implementation, will make the factorized form of the unitary coupled-cluster
approach much more efficient to implement on all types of quantum computers.
