  Much research regarding quantum adiabatic optimization has focused on
stoquastic Hamiltonians with Hamming symmetric potentials, such as the well
studied "spike" example. Due to the large amount of symmetry in these
potentials such problems are readily open to analysis both analytically and
computationally. However, more realistic potentials do not have such a high
degree of symmetry and may have many local minima. Here we present a somewhat
more realistic class of problems consisting of many individually Hamming
symmetric potential wells. For two or three such wells we demonstrate that such
a problem can be solved exactly in time polynomial in the number of qubits and
wells. For greater than three wells, we present a tight binding approach with
which to efficiently analyze the performance of such Hamiltonians in an
adiabatic computation. We provide several basic examples designed to highlight
the usefulness of this toy model and to give insight into using the tight
binding approach to examining it, including: (1) adiabatic unstructured search
with a transverse field driver and a prior guess to the marked item and (2) a
scheme for adiabatically simulating the ground states of small collections of
strongly interacting spins, with an explicit demonstration for an Ising model
Hamiltonian.
