  We study the quantum version of a simplified model of optimization problems,
where quantum fluctuations are introduced by a transverse field acting on the
qubits. We find a complex low-energy spectrum of the quantum Hamiltonian,
characterized by an abrupt condensation transition and a continuum of level
crossings as a function of the transverse field. We expect this complex
structure to have deep consequences on the behavior of quantum algorithms
attempting to find solutions to these problems.
