Using 1-Factorization from Graph Theory for Quantum Speedups on Clique Problems

TL;DR

This paper introduces quantum oracle designs based on 1-factorization of complete graphs, achieving faster quantum algorithms for clique problems and triangle finding, with improved depth and complexity over previous methods.

Contribution

It presents novel quantum oracle constructions using 1-factorization, reducing depth to O(n) and improving quantum algorithm efficiency for clique problems and triangle finding.

Findings

Quantum oracles with depth O(n) for clique problems

Reduced triangle finding complexity to O(n^{2.25} poly(log n))

Benchmarking of Amplitude Amplification iterations for k-CLIQUE

Abstract

The clique problems, including $k$-CLIQUE and Triangle Finding, form an important class of computational problems; the former is an NP-complete problem, while the latter directly gives lower bounds for Matrix Multiplication. A number of previous efforts have approached these problems with Quantum Computing methods, such as Amplitude Amplification. In this paper, we provide new Quantum oracle designs based on the 1-factorization of complete graphs, all of which have depth $O(n)$ instead of the $O(n^2)$ presented in previous studies. Also, we discuss the usage of one of these oracles in bringing the Triangle Finding time complexity down to $O(n^{2.25} poly(log n))$, compared to the $O(n^{2.38})$ classical record. Finally, we benchmark the number of required Amplitude Amplification iterations for another presented oracle, for solving $k$-CLIQUE.