  We study transport within a spatially heterogeneous one-dimensional quantum
walk with a combination of hierarchical and random barriers. Recent
renormalization group calculations for a spatially disordered quantum walk with
a regular hierarchy of barriers alone have shown a gradual decrease in
transport but no localization for increasing (but finite) barrier sizes. In
turn, it is well-known that extensive random disorder in the spatial barriers
is sufficient to localize a quantum walk on the line. Here we show that adding
only a sparse (sub-extensive) amount of randomness to a hierarchy of barriers
is sufficient to induce localization such that transport ceases. Our numerical
results suggest the existence of a localization transition for a combination of
both, the strength of the regular barrier hierarchy at large enough randomness
as well as the increasing randomness at sufficiently strong barriers in the
hierarchy.
