  The quantum approximate optimization algorithm (QAOA) is a promising method
for solving certain classical combinatorial optimization problems on near-term
quantum devices. When employing the QAOA to 3-SAT and Max-3-SAT problems, the
quantum cost exhibits an easy-hard-easy or easy-hard pattern respectively as
the clause density is changed. The quantum resources needed in the hard-region
problems are out of reach for current NISQ devices. We show by numerical
simulations with up to 14 variables and analytical arguments that the
adaptive-bias QAOA (ab-QAOA) greatly improves performance in the hard region of
the 3-SAT problems and hard region of the Max-3-SAT problems. For similar
accuracy, on average, ab-QAOA needs 3 levels for 10-variable 3-SAT problems as
compared to 22 for QAOA. For 10-variable Max-3-SAT problems, the numbers are 7
levels and 62 levels. The improvement comes from a more targeted and more
limited generation of entanglement during the evolution. We demonstrate that
classical optimization is not strictly necessary in the ab-QAOA since local
fields are used to guide the evolution. This leads us to propose an
optimization-free ab-QAOA that can solve the hard-region 3-SAT and Max-3-SAT
problems effectively with significantly fewer quantum gates as compared to the
original ab-QAOA. Our work paves the way for realizing quantum advantages for
optimization problems on NISQ devices.
