  The realization of robust universal quantum computation with any platform
ultimately requires both the coherent storage of quantum information and (at
least) one entangling operation between individual elements. The use of
continuous-variable bosonic modes as the quantum element is a promising route
to preserve the coherence of quantum information against naturally-occurring
errors. However, operations between bosonic modes can be challenging. In
analogy to the exchange interaction between discrete-variable spin systems, the
exponential-SWAP unitary [$\mathbf{U}_{\mathrm{E}}\left(\theta_c\right)$] can
coherently transfer the states between two bosonic modes, regardless of the
chosen encoding, realizing a deterministic entangling operation for certain
$\theta_c$. Here, we develop an efficient circuit to implement
$\mathbf{U}_{\mathrm{E}}\left(\theta_c\right)$ and realize the operation in a
three-dimensional circuit QED architecture. We demonstrate high-quality
deterministic entanglement between two cavity modes with several different
encodings. Our results provide a crucial primitive necessary for universal
quantum computation using bosonic modes.
