We study the Hamiltonian associated with the quantum adiabatic algorithm with
a random cost function. Because the cost function lacks structure we can prove
results about the ground state. We find the ground state energy as the number
of bits goes to infinity, show that the minimum gap goes to zero exponentially
quickly, and we see a localization transition. We prove that there are no
levels approaching the ground state near the end of the evolution. We do not
know which features of this model are shared by a quantum adiabatic algorithm
applied to random instances of satisfiability since despite being random they
do have bit structure
