Faster quantum walk search on a weighted graph
Thomas G. Wong · Physical Review A · 2015
A randomly walking quantum particle evolving by Schr\"odinger's equation searches for a unique marked vertex on the ``simplex of complete graphs'' in time $\mathrm{\ensuremath{\Theta}}({N}^{3/4})$. 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 $\mathrm{\ensuremath{\Theta}}(\sqrt{N})$. 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.