  Teleportation is a fundamental concept of quantum mechanics with an important
application in extending the range of quantum communication channels via
quantum relay nodes. To be compatible with real-world technology such as secure
quantum key distribution over fibre networks, such a relay node must operate at
GHz clock rates and accept time-bin encoded qubits in the low-loss telecom band
around 1550 nm. Here, we show that InAs/InP droplet epitaxy quantum dots with
their sub-Poissonian emission near 1550 nm are ideally suited for the
realisation of this technology. To create the necessary on-demand photon
emission at GHz clock rates, we develop a flexible pulsed optical excitation
scheme, and demonstrate that the fast driving conditions are compatible with a
low multiphoton emission rate. We show further that, even under these driving
conditions, photon pairs obtained from the biexciton cascade show an
entanglement fidelity close to 90\%, comparable to the value obtained under cw
excitation. Using asymetric Mach Zehnder interferometers and our photon source,
we finally construct a time-bin qubit quantum relay able to receive and send
time-bin encoded photons, and demonstrate mean teleportation fidelities of
$0.82\pm0.01$, exceeding the classical limit by nearly 10 standard deviations.
