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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 with relatively low-depth circuits, 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 scale 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 <span style="font-variant: small-caps;">Prepare</span> subroutine uses <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></semantics></math></inline-formula> ancilla qubits for a rank-<i>n</i> operator. Our <span style="font-variant: small-caps;">Select</span>(<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>U</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula>) scheme uses <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula><span style="font-variant: small-caps;">Cnot</span> gates. This results in a full algorithm that scales like the cube of the rank of the operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>n</mi><mn>3</mn></msup></semantics></math></inline-formula>, 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.