Cost vector analysis & multi-path entanglement routing in quantum networks

We present a static framework for analysing quantum routing protocols that we call the \textit{cost-vector formalism}. Here, quantum networks are recast as multi-graphs where edges represent two-qubit entanglement resources that \textit{could} exist under some sequence of operations. Each edge is weighted with a \textit{transmission probability} that represents the likelihood of the pair existing and a \textit{coherence probability} which is the likelihood that the pair is suitable for teleportation. Routing operations such as entanglement swapping and purification are then interpreted as \textit{contractions on the multi-graph} with relatively simple rules for updating the edge-weights. Moreover, we extend our formalism to include routing scenarios over time by developing a compatible resource theory for quantum memories. We develop rudimentary greedy algorithms for routing in this framework and test them over a variety of toy networking scenarios. Our results indicate that congestion in quantum networks does not improve significantly when more nodes (computers) are added. Rather, we find that congestion is all but eliminated by waiting a small amount of time.