Finding cliques by quantum adiabatic evolution

Quantum adiabatic evolution provides a general technique for the solution of
combinatorial search problems on quantum computers. We present the results of a
numerical study of a particular application of quantum adiabatic evolution, the
problem of finding the largest clique in a random graph. An n-vertex random
graph has each edge included with probability 1/2, and a clique is a completely
connected subgraph. There is no known classical algorithm that finds the
largest clique in a random graph with high probability and runs in a time
polynomial in n. For the small graphs we are able to investigate (n <= 18), the
quantum algorithm appears to require only a quadratic run time.