  Quantum networks are a new paradigm of complex networks, allowing us to
harness networked quantum technologies and to develop a quantum internet. But
how robust is a quantum network when its links and nodes start failing? We show
that quantum networks based on typical noisy quantum-repeater nodes are prone
to discontinuous phase transitions with respect to the random loss of operating
links and nodes, abruptly compromising the connectivity of the network, and
thus significantly limiting the reach of its operation. Furthermore, we
determine the critical quantum-repeater efficiency necessary to avoid this
catastrophic loss of connectivity as a function of the network topology, the
network size, and the distribution of entanglement in the network. In
particular, our results indicate that a scale-free topology is a crucial design
principle to establish a robust large-scale quantum internet.
