  Adiabatic quantum computation (AQC) is a promising counterpart of universal
quantum computation, based on the key concept of quantum annealing (QA). QA is
claimed to be at the basis of commercial quantum computers and benefits from
the fact that the detrimental role of decoherence and dephasing seems to have
poor impact on the annealing towards the ground state. While many papers show
interesting optimization results with a sizable number of qubits, a clear
evidence of a full quantum coherent behavior during the whole annealing
procedure is still lacking. In this paper we show that quantum non-demolition
(weak) measurements of Leggett Garg inequalities can be used to efficiently
assess the quantumness of the QA procedure. Numerical simulations based on a
weak coupling Lindblad approach are compared with classical Langevin
simulations to support our statements.
