Adiabatic Computation - A Toy Model

We discuss a toy model for adiabatic quantum computation which displays some phenomenological properties expected in more realistic implementations. This model has two free parameters: the adiabatic evolution parameter $s$ and the $\alpha$ parameter which emulates many-variables constrains in the classical computational problem. The proposed model presents, in the $s-\alpha$ plane, a line of first order quantum phase transition that ends at a second order point. The relation between computation complexity and the occurrence of quantum phase transitions is discussed. We analyze the behavior of the ground and first excited states near the quantum phase transition, the gap and the entanglement content of the ground state.