Entanglement Routing Based on Fidelity Curves for Quantum Photonics Channels

The quantum internet promises to extend entanglement correlations from nearby neighbors to any two nodes in a network. How to efficiently distribute entanglement over large-scale networks is still an open problem that greatly depends on the technology considered. In this work, we consider quantum networks composed of photonic channels characterized by a trade-off between the entanglement generation rate and fidelity. For such networks we look at the two following problems: the one of finding the best path to connect any two given nodes in the network bipartite entanglement routing, and the problem of finding the best starting node in order to connect three nodes in the network multipartite entanglement routing. We consider two entanglement distribution models: one where entangled qubit are distributed one at a time, and a flow model where a large number of entangled qubits are distributed simultaneously. We propose the use of continuous fidelity curves (i.e., entanglement generation fidelity vs rate) as the main routing metric. Combined with multi-objective path-finding algorithms, the fidelity curves describing each link allow finding a set of paths that maximize both the end-to-end fidelity and the entanglement generation rate. For the models and networks considered, we prove that the algorithm always converges to the optimal solution, and we show through simulation that its execution time grows polynomial with the number of nodes in the network. Our implementation grows with the number of nodes with a power between $1$ and $1.4$ depending on the network. This work paves the way for the development of path-finding algorithms for networks with complex entanglement distribution protocols, in particular for other protocols that exhibit a trade-off between generation fidelity and rate, such as repeater-and-purify protocols.