Faster quantum walk search on a weighted graph

A randomly walking quantum particle evolving by Schrödinger's equation searches for a unique marked vertex on the "simplex of complete graphs" in time Θ(N3/4). We give a weighted version of this graph that preserves vertex transitivity, and we show that the time to search on it can be reduced to nearly Θ(N). To prove this, we introduce two extensions to degenerate perturbation theory: an adjustment that distinguishes the weights of the edges and a method to determine how precisely the jumping rate of the quantum walk must be chosen.