mechanism has been tested in a variety of quantum toy models at zero temperature, including ordered and disordered systems, as well as for crossing isolated or extended critical regions (see, e.g., Ref. [@Dziarmaga_rev] for a review). When the annealing velocity is progressively increased, a crossover behaviour sets in between the KZ scaling and the generation of excitations due to faster quenches, where the dynamics can be described by the underlying classical model [@Silvi_2016].

The proliferation of defects due to Landau-Zener transitions is intimately related to the occurrence of errors in the AQC. While, in the unitary case, the number of defects decreases on increasing annealing time, the environment will be dominant for long annealing times. In this regime, one expects a defect formation which is almost independent on the annealing protocol. This picture was confirmed and detailed in Ref. [@Patane_2008], where the scaling in the crossover between the KZ-dominated regime and the environment-dominated regime was also found. In the presence of spatially correlated noise, additional intermediate regimes emerge due to the comparison of the correlation length of the noise and the correlation length of the system [@Nalbach_2015].

The role of temperature and external noise was further considered in the context 