  Ternary quantum information processing in superconducting devices poses a
promising alternative to its more popular binary counterpart through larger,
more connected computational spaces and proposed advantages in quantum
simulation and error correction. Although generally operated as qubits,
transmons have readily addressable higher levels, making them natural
candidates for operation as quantum three-level systems (qutrits). Recent works
in transmon devices have realized high fidelity single qutrit operation.
Nonetheless, effectively engineering a high-fidelity two-qutrit entanglement
remains a central challenge for realizing qutrit processing in a transmon
device. In this work, we apply the differential AC Stark shift to implement a
flexible, microwave-activated, and dynamic cross-Kerr entanglement between two
fixed-frequency transmon qutrits, expanding on work performed for the $ZZ$
interaction with transmon qubits. We then use this interaction to engineer
efficient, high-fidelity qutrit CZ$^\dag$ and CZ gates, with estimated process
fidelities of 97.3(1)% and 95.2(3)% respectively, a significant step forward
for operating qutrits on a multi-transmon device.
