Entanglement Routing and Bottlenecks in Grid Networks

Distributing entangled states among users is a fundamental problem in quantum networks. Existing protocols like $X$ protocol introduced in [npj Quantum Inf. 2019, 5, 76] use graph theoretic tools like local complementation to optimize the number of measurements required to extract any Bell pair among the network users. Such a protocol relies on finding the shortest path between the users. Here, it is established that, in general, the most optimal path to perform the $X$ protocol is not along the shortest path. Specific examples of this advantage are provided on networks of size as small as 12 qubits. Bottlenecks in establishing simultaneous Bell pairs in nearest-neighbor architectures are also explored. Recent results suggesting the unsuitability of the line and ring networks for the implementation of quantum networks due to the bottlenecks are revisited, and using local equivalency relations from graph theory, it's hinted that even grid graphs are not exempted from bottleneck issues. Finally, bottlenecks are simulated in grid graphs of sizes up to $6\times 6$. Analysis reveals relative positions within a grid graph that are more susceptible to bottlenecks. It demonstrates the superiority of the $X$ protocol along paths that are not shortest paths while avoiding bottlenecks for simultaneous communication requests.