We consider long-range quantum communication using quantum repeaters. We show
that by employing hashing, an entanglement distillation protocol with one-way
communication, one obtains a scalable scheme that allows one to reach arbitrary
distances, with constant overhead in resources per repeater station, and
ultrafast rates. This combines the advantages of direct encoded transmission
and entanglement-based repeaters, and reduces the required overhead per
transmitted qubit by a polynomial factor. In practical terms, we show that also
with moderate resources of a few hundred qubits at each repeater station, one
can reach intercontinental distances. At the same time, error thresholds for a
measurement-based implementation are of the order of several percent per qubit.
This opens the way for long-distance communication of big quantum data.