  We study the random transverse field Ising model on a finite Cayley tree.
This enables us to probe key questions arising in other important disordered
quantum systems, in particular the Anderson transition and the problem of dirty
bosons on the Cayley tree, or the emergence of non-ergodic properties in such
systems. We numerically investigate this problem building on the cavity
mean-field method complemented by state-of-the art finite-size scaling
analysis. Our numerics agree very well with analytical results based on an
analogy with the traveling wave problem of a branching random walk in the
presence of an absorbing wall. Critical properties and finite-size corrections
for the zero-temperature paramagnetic-ferromagnetic transition are studied both
for constant and algebraically vanishing boundary conditions. In the later
case, we reveal a regime which is reminiscent of the non-ergodic delocalized
phase observed in other systems, thus shedding some light on critical issues in
the context of disordered quantum systems, such as Anderson transitions, the
many-body localization or disordered bosons in infinite dimensions.
