[H(t), \rho] + \mathbb{D}[\rho] .
  \label{eq:master}$$ The first term on the right-hand side describes the coherent unitary time evolution, which is ruled by a many-body time-varying Hamiltonian $H(t)$, according to the quantum annealing protocol . The second term on the right-hand side of Eq.  accounts for the coupling to the environment, and its form will depend on the nature of noise and dissipation as well as the form of the coupling between the many-body system and the external bath. The dissipator is a completely positive, trace preserving map, that – in general – drives the system into a fixed point, that is a steady state or a steady-state manifold (if the fixed point is not unique). As such, it is inducing a decay in the system towards the steady state. Note that the differential form of Eq.  generally requires a Markovian bath. Understanding the effect of dissipation on AQC amounts to quantifying, in some way, how the state of the system deviates from the final ground state, because of the presence of the extra term $\mathbb{D}[\rho]$ in Eq. .

Does the presence of the environment facilitate the reaching of the final ground state or is it detrimental for AQC? It 