  We give a quantum algorithm for solving instances of the satisfiability
problem, based on adiabatic evolution. The evolution of the quantum state is
governed by a time-dependent Hamiltonian that interpolates between an initial
Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian,
whose ground state encodes the satisfying assignment. To ensure that the system
evolves to the desired final ground state, the evolution time must be big
enough. The time required depends on the minimum energy difference between the
two lowest states of the interpolating Hamiltonian. We are unable to estimate
this gap in general. We give some special symmetric cases of the satisfiability
problem where the symmetry allows us to estimate the gap and we show that, in
these cases, our algorithm runs in polynomial time.
