  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.
