Connectivity is a Poor Indicator of Fast Quantum Search

TL;DR

This paper demonstrates that graph connectivity is not a reliable indicator of quantum search efficiency by providing counterexamples with contrasting connectivity and search times, including a novel two-stage quantum walk algorithm.

Contribution

The paper challenges the assumption that high connectivity implies fast quantum search by presenting counterexamples and introducing a new two-stage quantum walk method.

Findings

Low connectivity graph with fast search

High connectivity graph with slow search

Introduction of a two-stage quantum walk algorithm

Abstract

A randomly walking quantum particle evolving by Schr\"odinger's equation searches on $d$-dimensional cubic lattices in $O(\sqrt{N})$ time when $d \ge 5$, and with progressively slower runtime as $d$ decreases. This suggests that graph connectivity (including vertex, edge, algebraic, and normalized algebraic connectivities) is an indicator of fast quantum search, a belief supported by fast quantum search on complete graphs, strongly regular graphs, and hypercubes, all of which are highly connected. In this paper, we show this intuition to be false by giving two examples of graphs for which the opposite holds true: one with low connectivity but fast search, and one with high connectivity but slow search. The second example is a novel two-stage quantum walk algorithm in which the walking rate must be adjusted to yield high search probability.