  Resonators with weak single-photon self-Kerr nonlinearities can theoretically
be used to prepare Fock states in the presence of a loss much larger than their
nonlinearities. Two necessary ingredients are large displacements and a
two-photon (parametric) drive. Here, we find that these systems can be
controlled to achieve any desired gate operation in a finite dimensional
subspace (whose dimensionality can be chosen at will). Moreover, we show that
the two-photon driving requirement can be relaxed and that full controllability
is achievable with only 1-photon (linear) drives. We make use of both
Trotter-Suzuki decompositions and gradient-based optimization to find control
pulses for a desired gate, which reduces the computational overhead by using a
small blockaded subspace. We also discuss the infidelity arising from input
power limitations in realistic settings, as well as from corrections to the
rotating-wave approximation. Our universal control protocol opens the
possibility for quantum information processing using a wide range of lossy
systems with weak nonlinearities.
