  We introduce a new scheme for quantum circuit design called controlled gate
networks. Rather than trying to reduce the complexity of individual unitary
operations, the new strategy is to toggle between all of the unitary operations
needed with the fewest number of gates. We illustrate our approach using two
examples. The first example is a variational subspace calculation for a
two-qubit system. We demonstrate an approximately five-fold reduction in the
number of two-qubit gates required for computing inner products and Hamiltonian
matrix elements. The second example is estimating the eigenvalues of a
two-qubit Hamiltonian via the Rodeo Algorithm using a specific class of
controlled gate networks called controlled reversal gates. Again, a fivefold
reduction in the number of two-qubit gates is demonstrated. We use the
Quantinuum H1-2 and IBM Perth devices to realize the quantum circuits. Our work
demonstrates that controlled gate networks are a useful tool for reducing gate
complexity in quantum algorithms for quantum many-body problems.
