Quantum speed-up in solving the maximal clique problem

TL;DR

This paper presents a quantum algorithm that achieves quadratic speed-up for solving the NP-complete maximal clique problem, demonstrated through a nuclear magnetic resonance experiment on a small graph.

Contribution

It introduces an optimal quantum algorithm for the maximal clique problem with reduced time and space complexity, validated by experimental implementation.

Findings

Quantum algorithm reduces complexity to O(√2^n)

Experimental NMR implementation on a 2-vertex graph

Algorithm is optimal among oracle-based quantum solutions

Abstract

The maximal clique problem, to find the maximally sized clique in a given graph, is classically an NP-complete computational problem, which has potential applications ranging from electrical engineering, computational chemistry, bioinformatics to social networks. Here we develop a quantum algorithm to solve the maximal clique problem for any graph $G$ with $n$ vertices with quadratic speed-up over its classical counterparts, where the time and spatial complexities are reduced to, respectively, $O(\sqrt{2^{n}})$ and $O(n^{2})$. With respect to oracle-related quantum algorithms for the NP-complete problems, we identify our algorithm to be optimal. To justify the feasibility of the proposed quantum algorithm, we have successfully solved an exemplified clique problem for a graph $G$ with two vertices and one edge by carrying out a nuclear magnetic resonance experiment involving four qubits.