field of quantum technologies have drastically enhanced our capability to control the quantum coherent dynamics of many-body systems in a variety of physical systems, ranging from atomic and molecular optics, to trapped ions, and cavity/circuit quantum electrodynamics. These progresses have made real the possibility to experimentally realise quantum simulators [@NatPhys_2012] as well as the implementation of the first quantum algorithms [@Corcoles_2015; @Chow_2014].

Together with the progresses in implementing quantum gates and concatenate them, i.e., by realising standard circuit computation [@NielsenChuang], recently adiabatic quantum computation (AQC) [@Fahri_2001] and quantum annealing [@Santoro_2002] have received a tremendous boost thanks to the experiments performed with D-Wave machines [@Denchev_2016; @Lanting_2014; @King_2016; @Venturelli_2015]. The strategy underlying adiabatic quantum computation [@Fahri_2001; @Santoro_2002] is based on the fact that any quantum algorithm can be formulated in terms of identifying the global minimum (ground state) of a given function (Hamiltonian) over a set of many local minima. On the experimental side, quantum effects were seen to survive on eight [@Johnson_2011; @Boixo_2013], sixteen [@Dickson_2013] and even in more than one hundred qubits [@Boixo_2014]. Whether these machines hold already at present the so called “quantum supremacy” or not is still under debate [@Ronnow_2014; @Boixo_2016]. However, it is clearly very important to 