Designing parametrically-driven qudit gates in a modular SNAIL architecture

Quantum circuits are typically described using qubit-based interactions, but research into qudit (d-level system)-based circuits can be useful for quantum simulation, compressed compilation, and error correction. A modular architecture with a tunable SNAIL provides the ability to drive these multi-level interactions between qudits [Zhou, et al. npj Quantum Information (2023)]. Here we engineer single-qudit and two-qudit gates with parametric processes on the SNAIL modular architecture. Understanding qudit-based interactions requires the exploration of parametric processes that traverse multiple excitation levels; however, the order of the SNAIL nonlinearities limits the parametric interactions that can be driven. Concretely, we build various implementations of a CX gate between non-adjacent levels, comparing a 5^th order coupling with multiple simultaneous 3^rd order couplings. We characterize their fidelities and error channels, which is crucial for leveraging these interactions in future quantum systems.