Algorithms for finding the maximum clique based on continuous time quantum walks

Abstract

In this work, the application of continuous time quantum walks (CTQW) to the Maximum Clique (MC) problem was studied. Performing CTQW on graphs can generate distinct periodic probability amplitudes for different vertices. We found that the intensities of the probability amplitudes at some frequencies imply the clique structure of special kinds of graphs. Recursive algorithms with time complexity $O(N^6)$ in classical computers were proposed to determine the maximum clique. We have experimented on random graphs where each edge exists with different probabilities. Although counter examples were not found for random graphs, whether these algorithms are universal is beyond the scope of this work.

A parameter-independent algorithm of finding maximum clique with Seidel continuous-time quantum walks

“…Unlike other quantum algorithms, quantum walks aim to solve problems by obtaining a desirable probability distribution 27 , 28 , 29 , 30 , 31 , 32 , 33 or comparing the probability amplitudes of two walking systems. 34 , 35 In this study, we investigate the probability amplitudes of continuous-time quantum walks (CTQW) on several special graphs and use these amplitudes to deduce a clique within the given graph. In the initial version of CTQW, the adjacency matrix, or Laplacian matrix of the graph is commonly used as the driving Hamiltonian.…”