of AQC in several papers, showing that in some cases it may be beneficial in reaching the target ground state. Work has been done on the comparison between AQC and classical approaches using thermal hopping [@Boixo_2016b], on the use of quantum diffusion, showing better performance than closed-system quantum annealing [@Smelyanskiy_2017] as well as on the crucial role of noise-induced thermalisation in AQC that can outperform simulated annealing [@Kechedzhi_2016]. Further, the relation of thermally-assisted tunneling to quantum Monte Carlo has been studied in Ref. [@Jiang_2017]. In addition to this, the effect of noise of a thermal environment [@Albash_2015] and decoherence [@Albash_2012] have been studied as well.

It should be kept in mind that the adiabatic dynamics of dissipative many-body system is linked to the understanding of the Landau-Zener problem of a two-level system coupled to an environment, an extensive studied problem in many different areas [@Ao_1989; @Wubs_2006; @Ashhab_2006; @Zueco_2012; @Javanbakht_2015; @Arceci_2017].

The problem of describing the adiabatic dynamics of a many-body open quantum system is a formidable problem and approximations are necessary. It is however of fundamental importance to have some non-trivial examples where the outcomes of the analysis are not hampered by any approximation. The aim of this work is 