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The quantum internet promises to extend entanglement correlations from nearbyneighbors to any two nodes in a network. How to efficiently distributeentanglement over large-scale networks is still an open problem that greatlydepends on the technology considered. In this work, we consider quantumnetworks composed of photonic channels characterized by a trade-off between theentanglement generation rate and fidelity. For such networks we look at the twofollowing problems: the one of finding the best path to connect any two givennodes in the network bipartite entanglement routing, and the problem of findingthe best starting node in order to connect three nodes in the networkmultipartite entanglement routing. We consider two entanglement distributionmodels: one where entangled qubit are distributed one at a time, and a flowmodel where a large number of entangled qubits are distributed simultaneously.We propose the use of continuous fidelity curves (i.e., entanglement generationfidelity vs rate) as the main routing metric. Combined with multi-objectivepath-finding algorithms, the fidelity curves describing each link allow findinga set of paths that maximize both the end-to-end fidelity and the entanglementgeneration rate. For the models and networks considered, we prove that thealgorithm always converges to the optimal solution, and we show throughsimulation that its execution time grows polynomial with the number of nodes inthe network. Our implementation grows with the number of nodes with a powerbetween $1$ and $1.4$ depending on the network. This work paves the way for thedevelopment of path-finding algorithms for networks with complex entanglementdistribution protocols, in particular for other protocols that exhibit atrade-off between generation fidelity and rate, such as repeater-and-purifyprotocols.

Entanglement Routing Based on Fidelity Curves for Quantum Photonics Channels