We show that strong clustering of links in complex networks, i.e., a high probability of triadic closure, can induce a localization-delocalization quantum phase transition (Anderson-like transition) of coherent excitations. For example, the propagation of light wave packets between two distant nodes of an optical network (composed of fibers and beam splitters) will be absent if the fraction of closed triangles exceeds a certain threshold. We suggest that such an experiment is feasible with current optics technology. We determine the corresponding phase diagram as a function of clustering coefficient and disorder for scale-free networks of different degree distributions P(k) approximately k;{-lambda}. Without disorder, we observe no phase transition for lambda<4, a quantum transition for lambda>4, and an additional distinct classical transition for lambda>4.5. Disorder reduces the critical clustering coefficient such that phase transitions occur for smaller lambda.