We calculate the energy spectrum and eigenstates of a graphene sheet which contains a circular deformation. Using time-independent perturbation theory with the ratio of the height and width of the deformation as the small parameter, we find that due
to the curvature the wavefunctions for the various states acquire unique angular asymmetry. We demonstrate that the pseudo-magnetic fields induced by the curvature result in circulating probability currents. These circulating currents in turn produce local textit{real} magnetic fields $sim$ 100 $mu$T which can be measured using current technology.
We investigate an extended version of the quantum Ising model which includes beyond-nearest neighbour interactions and an additional site-dependent longitudinal magnetic field. Treating the interaction exactly and using perturbation theory in the lon
gitudinal field, we calculate the energy spectrum and find that the presence of beyond-nearest-neighbour interactions enhances the minimal gap between the ground state and the first excited state, irrespective of the nature of decay of these interactions along the chain. The longitudinal field adds a correction to this gap that is independent of the number of qubits. We discuss the application of our model to implementing specific instances of 3-satisfiability problems (Exact Cover) and make a connection to a chain of flux qubits.
