Quantum Algorithm for Finding a Maximum Clique in an Undirected Graph

Abstract

The maximum clique in an undirected graph is the largest subset of  a set of graph's vertices where each pair of elements in the subset is connected. In this paper I would like to propose an algorithm for quantum computers that finds a maximum clique in an arbitrary undirected graph. An algorithm has O(|V|2^|V|/2) worst case time complexity and O(2^|V|/2) best case time complexity. Algorithm's space complexity for each case is O(|V|). An algorithm is based on a version of Grover's  quantum algorithm for finding element in an unsorted list in which there can be an unknown number of solutions.