A quantum searching model finding one of the edges of a subgraph in a complete graph

TL;DR

This paper introduces a quantum search model that efficiently finds an edge of a subgraph within a complete graph using perturbed quantum walks, achieving quadratic speed-up over classical methods.

Contribution

We extend previous work by demonstrating that the quantum search model is effective for any subgraph, not just matchings, through spectral analysis.

Findings

Quadratic speed-up in finding subgraph edges

Model applicable to any subgraph in a complete graph

Spectral analysis confirms efficiency

Abstract

Some of the quantum searching models have been given by perturbed quantum walks. Driving some perturbed quantum walks, we may quickly find one of the targets with high probability. In this paper, we construct a quantum searching model finding one of the edges of a given subgraph in a complete graph. How to construct our model is that we label the arcs by $+1$ or $-1$, and define a perturbed quantum walk by the sign function on the set of arcs. After that, we detect one of the edges labeled $-1$ by the induced sign function as fast as possible. This idea was firstly proposed by Segawa et al. in 2021. They only addressed the case where the subgraph forms a matching, and obtained by a combinatorial argument that the time of finding one of the edges of the subgraph is quadratically faster than a classical searching model. In this paper, we show that the model is valid for any subgraph, i.e., we obtain by spectral analysis a quadratic speed-up for finding one of the edges of the subgraph in a complete graph.