A quantum differentiation of k-SAT instances

We present a quantum adiabatic algorithm to differentiate between k-SAT instances, those with no solutions and those that have many solutions. The time complexity of the algorithm is a function of the energy gap between the subspace of all O-eigenvectors (ground states) and the first excited states manifold, and scales polynomially with the number of resources. The idea of gaps between subspaces suggests a new tool to analyze time complexity in adiabatic quantum machines.