  Topological qubits based on Majorana fermions have the potential to
revolutionize the emerging field of quantum computing by making information
processing significantly more robust to decoherence. Nanowires (NWs) are a
promising medium for hosting these kinds of qubits, though branched NWs are
needed to perform qubit manipulations. Here we report gold-free templated
growth of III-V NWs by molecular beam epitaxy using an approach that enables
patternable and highly regular branched NW arrays on a far greater scale than
what has been reported thus far. Our approach relies on the lattice-mismatched
growth of InAs on top of defect-free GaAs nanomembranes (NMs) yielding
laterally-oriented, low-defect InAs and InGaAs NWs whose shapes are determined
by surface and strain energy minimization. By controlling NM width and growth
time, we demonstrate the formation of compositionally graded NWs with
cross-sections less than 50 nm. Scaling the NWs below 20 nm leads to the
formation of homogenous InGaAs NWs which exhibit phase-coherent, quasi-1D
quantum transport as shown by magnetoconductance measurements. These results
are an important advance towards scalable topological quantum computing.
