  A previously developed quantum search algorithm for solving 1-SAT problems in
a single step is generalized to apply to a range of highly constrained k-SAT
problems. We identify a bound on the number of clauses in satisfiability
problems for which the generalized algorithm can find a solution in a constant
number of steps as the number of variables increases. This performance
contrasts with the linear growth in the number of steps required by the best
classical algorithms, and the exponential number required by classical and
quantum methods that ignore the problem structure. In some cases, the algorithm
can also guarantee that insoluble problems in fact have no solutions, unlike
previously proposed quantum search algorithms.
