  We present the results of a numerical study, with 20 qubits, of the
performance of the Quantum Adiabatic Algorithm on randomly generated instances
of MAX 2-SAT with a unique assignment that maximizes the number of satisfied
clauses. The probability of obtaining this assignment at the end of the quantum
evolution measures the success of the algorithm. Here we report three
strategies which consistently increase the success probability for the hardest
instances in our ensemble: decreasing the overall evolution time, initializing
the system in excited states, and adding a random local Hamiltonian to the
middle of the evolution.
