  In continuous-variable quantum computing with qubits encoded in the
infinite-dimensional Hilbert space of bosonic modes, it is a difficult task to
realize strong and on-demand interactions between the qubits. One option is to
engineer a beamsplitter interaction for photons in two superconducting cavities
by driving an intermediate superconducting circuit with two continuous-wave
drives, as demonstrated in a recent experiment. Here, we show how quantum
optimal control theory (OCT) can be used in a systematic way to improve the
beamsplitter interaction between the two cavities. We find that replacing the
two-tone protocol by a three-tone protocol accelerates the effective
beamsplitter rate between the two cavities. The third tone's amplitude and
frequency are determined by gradient-free optimization and make use of
cavity-transmon sideband couplings. We show how to further improve the
three-tone protocol via gradient-based optimization while keeping the optimized
drives experimentally feasible. Our work exemplifies how to use OCT to
systematically improve practical protocols in quantum information applications.
