understand the actual mode of operation of these adiabatic computers, in order to understand the limit of their performances and to push it forward.

A key problem in this framework is to understand the role of dissipation and decoherence on adiabatic quantum computers. This question amounts to understanding the key features that control the adiabatic evolution of a many-body open quantum system.

Let us first state the general problem. Suppose to be able to follow the quantum dynamics of an appropriate time-dependent Hamiltonian: $$H(t) = [1-f(t)] \, H_{\rm in} + f(t) \, H_{\rm fin}, \quad t \in [t_{\rm in}, t_{\rm fin}],
  \label{eq:QAnnealing}$$ $f(t)$ being a generic function of time, with $f(t_{\rm in})=0$ and $f(t_{\rm fin}) = 1$. The Hamiltonian $H_{\rm in}$ sets the initial condition as its ground state, while the sought solution to the problem is entailed into the ground state of $H_{\rm fin}$. If the control time is much larger than the typical inverse gap between the ground state and the first excited state, the system will adiabatically follow its instantaneous ground state $|\psi_0(t)\rangle$. Reaching the ground state of $H_{\rm fin}$ by adiabatic evolution is the way AQC works [@Fahri_2001; @Santoro_2002]. As long as the evolution is unitary, 