A controlled-SWAP operation between two bosonic modes

The coherence and large Hilbert spaces of three-dimensional superconducting microwave cavities make them an attractive platform for building quantum memories. Universal control of a single cavity mode can be accomplished by dispersively-coupling the cavity to a nonlinear mode such as a transmon. Wedding this resource with a bilinear cavity-cavity coupling such as a beamsplitter interaction provides a modular framework for quantum control of multiple cavities. Here we discuss a demonstration of this approach in the form of a Fredkin gate (controlled-SWAP) between a pair of cavities coupled by a three-wave mixing element, in which a transmon provides the control bit for the SWAP.