  Quantum computing based on Neutral Atoms (NAs) provides a wide range of
computational capabilities, encompassing high-fidelity long-range interactions
with native multi-qubit gates, and the ability to shuttle arrays of qubits.
While previously these capabilities have been studied individually, we propose
the first approach of a fast hybrid compiler to perform circuit mapping and
routing based on both high-fidelity gate interactions and qubit shuttling. We
delve into the intricacies of the compilation process when combining multiple
capabilities and present effective solutions to address resulting challenges.
The final compilation strategy is then showcased across various hardware
settings, revealing its versatility, and highlighting potential fidelity
enhancements achieved through the strategic utilization of combined gate- and
shuttling-based routing. With the additional multi-qubit gate support for both
routing capabilities, the proposed approach is able to take advantage of the
full spectrum of computational capabilities offered by NAs.
