  Quantum spin-Hall edges are envisaged as next-generation transistors, yet
they exhibit dissipationless transport only over short distances. Here we show
that in a diffusive sample, where charge puddles with odd spin cause
back-scattering, a magnetic field drastically increases the mean free path and
drives the system into the ballistic regime with a Landauer-Buttiker
conductance. A strong non-linear non-reciprocal current emerges in the
diffusive regime with opposite signs on each edge, and vanishes in the
ballistic limit. We discuss its detection in state-of-the-art experiments.
