  We present a new approach to quantum computation involving the geometric
phase. In this approach, an entire computation is performed by adiabatically
evolving a suitably chosen quantum system in a closed circuit in parameter
space. The problem solved is the determination of the solubility of a 3-SAT
Boolean Satisfiability problem. The problem of non-adiabatic transitions to
higher levels is addressed in several ways. The avoided level crossings are
well defined and the interpolation can be slowed in this region, the
Hamiltonian can be scaled with problem dimension resulting in a constant gap
size and location, and the prescription here is sufficiently general as to
allow for other suitably chosen Hamiltonians. Finally, we show that with $n$
applications of this approach, the geometric phase based quantum computation
method may be used to find the solution to the 3-SAT problem in $n$ variables,
a member of the NP-complete complexity class.
