  We investigate quantum transport in binary tree structures and in hypercubes
for the disordered Frenkel-exciton Hamiltonian under pure dephasing noise. We
compute the energy transport efficiency as a function of disorder and dephasing
rates. We demonstrate that dephasing improves transport efficiency not only in
the disordered case, but also in the ordered one. The maximal transport
efficiency is obtained when the dephasing timescale matches the hopping
timescale, which represent new examples of the Goldilocks principle at the
quantum scale. Remarkably, we find that in weak dephasing regimes, away from
optimal levels of environmental fluctuations, the average effect of increasing
disorder is to improve the transport efficiency until an optimal value for
disorder is reached. Our results suggest that rational design of the site
energies statistical distributions could lead to better performances in
transport systems at nanoscale when their natural environments are far from the
optimal dephasing regime.
