  We propose and analyze two types of microwave-activated gates between a
fluxonium and a transmon qubit, namely a cross-resonance (CR) and a CPHASE
gate. The large frequency difference between a transmon and a fluxonium makes
the realization of a two-qubit gate challenging. For a medium-frequency
fluxonium qubit, the transmon-fluxonium system allows for a cross-resonance
effect mediated by the higher levels of the fluxonium over a wide range of
transmon frequencies. This allows one to realize the cross-resonance gate by
driving the fluxonium at the transmon frequency, mitigating typical problems of
the cross-resonance gate in transmon-transmon chips related to frequency
targeting and residual ZZ coupling. However, when the fundamental frequency of
the fluxonium enters the low-frequency regime below 100 MHz, the
cross-resonance effect decreases leading to long gate times. For this range of
parameters, a fast microwave CPHASE gate can be implemented using the higher
levels of the fluxonium. In both cases, we perform numerical simulations of the
gate showing that a gate fidelity above 99% can be obtained with gate times
between 100 and 300 ns. Next to a detailed gate analysis, we perform a study of
chip yield for a surface code lattice of fluxonia and transmons interacting via
the proposed cross-resonance gate. We find a much better yield as compared to a
transmon-only architecture with the cross-resonance gate as native two-qubit
gate.
