  We introduce a new multimode cavity QED architecture for superconducting
circuits which can be used to implement photonic memories, more efficient
Purcell filters, and quantum simulations of photonic materials. We show that
qubit interactions mediated by multimode cavities can have exponentially
improved contrast for two qubit gates without sacrificing gate speed. Using
two-qubits coupled via a three-mode cavity system we spectroscopically observe
multimode strong couplings up to 102MHz and demonstrate suppressed interactions
off-resonance of 10kHz when the qubits are ~600MHz detuned from the cavity
resonance. We study Landau-Zener transitions in our multimode systems and
demonstrate quasi-adiabatic loading of single photons into the multimode cavity
in 25ns. We introduce an adiabatic gate protocol to realize a controlled-Z gate
between the qubits in 95ns and create a Bell state with 94.7% fidelity. This
corresponds to an on/off ratio (gate contrast) of 1000.
