Searching Weighted Barbell Graphs with Laplacian and Adjacency Quantum Walks

TL;DR

This paper investigates quantum search algorithms on weighted barbell graphs, revealing that adjacency walks can be optimized with a two-stage approach to significantly improve success probability, while Laplacian walks remain unaffected by edge weights.

Contribution

It introduces a two-stage quantum walk algorithm on weighted barbell graphs that enhances search success probability, demonstrating the impact of edge weights on adjacency but not Laplacian walks.

Findings

Laplacian quantum walk behavior is unaffected by bridge weight.

Adjacency quantum walk can be optimized with a two-stage algorithm.

Success probability increases from 0.5 to 0.996 with optimal weighting.

Abstract

A quantum particle evolving by Schr\"odinger's equation in discrete space constitutes a continuous-time quantum walk on a graph of vertices and edges. When a vertex is marked by an oracle, the quantum walk effects a quantum search algorithm. Previous investigations of this quantum search algorithm on graphs with cliques have shown that the edges between the cliques can be weighted to enhance the movement of probability between the cliques to reach the marked vertex. In this paper, we explore the most restrictive form of this by analyzing search on a weighted barbell graph that consists of two cliques of the same size joined by a single weighted edge/bridge. This graph is generally irregular, so quantum walks governed by the graph Laplacian or by the adjacency matrix can differ. We show that the Laplacian quantum walk's behavior does not change, no matter the weight of the bridge, and so the single bridge is too restrictive to affect the walk. Similarly, the adjacency quantum walk's behavior is unchanged for most weights, but when the weight equals the size of a clique, the probability does collect at the clique containing the marked vertex, and utilizing a two-stage algorithm with different weights for each stage, the success probability is boosted from 0.5 to 0.996, independent of the size of the barbell graph.