  We study the typical (median) value of the minimum gap in the quantum version
of the Exact Cover problem using Quantum Monte Carlo simulations, in order to
understand the complexity of the quantum adiabatic algorithm (QAA) for much
larger sizes than before. For a range of sizes, N <= 128, where the classical
Davis-Putnam algorithm shows exponential median complexity, the QAA shows
polynomial median complexity. The bottleneck of the algorithm is an isolated
avoided crossing point of a Landau-Zener type (collision between the two lowest
energy levels only).
