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Quantum networks provide access to exchange of quantum information. Theprimary task of quantum networks is to distribute entanglement between remotenodes. Although quantum repeater protocol enables long distance entanglementdistribution, it has been restricted to one-dimensional linear network. Here wedevelop a general framework that allows application of quantum repeaterprotocol to arbitrary quantum repeater networks with fractal structure.Entanglement distribution across such networks is mapped to renormalization.Furthermore, we demonstrate that logarithmical times of recursive suchrenormalization transformations can trigger fractal to small-world transition,where a scalable quantum small-world network is achieved. Our result providesnew insight into quantum repeater theory towards realistic construction oflarge-scale quantum networks.

Renormalization and small-world model of fractal quantum repeater networks