Faster Quantum Walk Search on a Weighted Graph

TL;DR

This paper improves quantum walk search efficiency on a weighted, vertex-transitive graph, reducing search time from /4 power to nearly square root of N by novel perturbation techniques.

Contribution

It introduces a weighted graph version of previous models and develops two new perturbation methods to optimize quantum search times.

Findings

Search time reduced to nearly /4 power of N

Weighted graph preserves vertex-transitivity

New perturbation techniques enable precise control of quantum walk

Abstract

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 $\Theta(N^{3/4})$. In this paper, 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 $\Theta(\sqrt{N})$. To prove this, we introduce two novel 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.