  In the continuum limit (large number of qubits), adiabatic quantum algorithms
display a remarkable similarity to sweeps through quantum phase transitions. We
find that transitions of second or higher order are advantageous in comparison
to those of first order. With this insight, we propose a novel adiabatic
quantum algorithm for the solution of 3-satisfiability (3-SAT) problems (exact
cover), which is significantly faster than previous proposals according to
numerical simulations (up to 20 qubits). These findings suggest that adiabatic
quantum algorithms can solve NP-complete problems such as 3-SAT much faster
than the Grover search routine (yielding a quadratic enhancement), possibly
even with an exponential speed-up. PACS: 03.67.-a, 03.67.Lx, 73.43.Nq,
64.70.-p.
