  In hybrid circuit QED architectures containing both ancilla qubits and
bosonic modes, a controlled beam splitter gate is a powerful resource. It can
be used to create (up to a controlled-parity operation) an ancilla-controlled
SWAP gate acting on two bosonic modes. This is the essential element required
to execute the `swap test' for purity, prepare quantum non-Gaussian
entanglement and directly measure nonlinear functionals of quantum states. It
also constitutes an important gate for hybrid discrete/continuous-variable
quantum computation. We propose a new realization of a hybrid cSWAP utilizing
`Kerr-cat' qubits -- anharmonic oscillators subject to strong two-photon
driving. The Kerr-cat is used to generate a controlled-phase beam splitter
(cPBS) operation. When combined with an ordinary beam splitter one obtains a
controlled beam-splitter (cBS) and from this a cSWAP. The strongly biased error
channel for the Kerr-cat has phase flips which dominate over bit flips. This
yields important benefits for the cSWAP gate which becomes non-destructive and
transparent to the dominate error. Our proposal is straightforward to implement
and, based on currently existing experimental parameters, should achieve
controlled beam-splitter gates with high fidelities comparable to current
ordinary beam-splitter operations available in circuit QED.
