  We study a quantum small-world network with disorder and show that the system
exhibits a delocalization transition. A quantum algorithm is built up which
simulates the evolution operator of the model in a polynomial number of gates
for exponential number of vertices in the network. The total computational gain
is shown to depend on the parameters of the network and a larger than quadratic
speed-up can be reached.
  We also investigate the robustness of the algorithm in presence of
imperfections.
