  Novel qubits with intrinsic noise protection constitute a promising route for
improving the coherence of quantum information in superconducting circuits.
However, many protected superconducting qubits exhibit relatively low
transition frequencies, which could make their integration with conventional
transmon circuits challenging. In this work, we propose and study a scheme for
entangling a tunable transmon with a Cooper-pair parity-protected qubit, a
paradigmatic example of a low-frequency protected qubit that stores quantum
information in opposite Cooper-pair parity states on a superconducting island.
By tuning the external flux on the transmon, we show that non-computational
states can mediate a two-qubit entangling gate that preserves the Cooper-pair
parity independent of the detailed pulse sequence. Interestingly, the
entangling gate bears similarities to a controlled-phase gate in conventional
transmon devices. Hence, our results suggest that standard high-precision gate
calibration protocols could be repurposed for operating hybrid qubit devices.
