  With the emergence of the Quantum Internet, the need for advanced quantum
networking techniques has significantly risen. Various models of quantum
repeaters have been presented, each delineating a unique strategy to ensure
quantum communication over long distances. We focus on repeaters that employ
entanglement generation and swapping. This revolves around establishing remote
end-to-end entanglement through repeaters, a concept we denote as the
"quantum-native" repeaters (also called "first-generation" repeaters in some
literature). The challenges in routing with quantum-native repeaters arise from
probabilistic entanglement generation and restricted coherence time. Current
approaches use synchronized time slots to search for entanglement-swapping
paths, resulting in inefficiencies. Here, we propose a new set of asynchronous
routing protocols for quantum networks by incorporating the idea of maintaining
a dynamic topology in a distributed manner, which has been extensively studied
in classical routing for lossy networks, such as using a destination-oriented
directed acyclic graph (DODAG) or a spanning tree. The protocols update the
entanglement-link topology asynchronously, identify optimal
entanglement-swapping paths, and preserve unused direct-link entanglements. Our
results indicate that asynchronous protocols achieve a larger upper bound with
an appropriate setting and significantly higher entanglement rate than existing
synchronous approaches, and the rate increases with coherence time, suggesting
that it will have a much more profound impact on quantum networks as technology
advances.
