Quantum Algorithm for Testing Graph Completeness

TL;DR

This paper introduces a quantum algorithm that efficiently tests graph completeness using quantum walks and phase estimation, achieving logarithmic squared time complexity and demonstrating quantum advantage over classical methods.

Contribution

It presents a novel quantum algorithm leveraging Szegedy quantum walk and QPE for testing graph completeness with improved efficiency.

Findings

Time complexity is $oxed{ ext{O}( ext{log}^2 n)}$ for large graphs.

Quantum approach outperforms classical algorithms in speed.

Applicable to network analysis and system evaluation.

Abstract

Testing graph completeness is a critical problem in computer science and network theory. Leveraging quantum computation, we present an efficient algorithm using the Szegedy quantum walk and quantum phase estimation (QPE). Our algorithm, which takes the number of nodes and the adjacency matrix as input, constructs a quantum walk operator and applies QPE to estimate its eigenvalues. These eigenvalues reveal the graph's structural properties, enabling us to determine its completeness. We establish a relationship between the number of nodes in a complete graph and the number of marked nodes, optimizing the success probability and running time. The time complexity of our algorithm is $\mathcal{O}(\log^2n)$, where $n$ is the number of nodes of the graph. offering a clear quantum advantage over classical methods. This approach is useful in network structure analysis, evaluating classical routing algorithms, and assessing systems based on pairwise comparisons.