the environment.

In the next sections we will first address analytically the fermionic model  with dissipation provided by Lindblad terms as in -, without making any reference to the mapping with spins. The effect of the non-local part of the JWT will be only discussed in Sec. \[sec:Ising\], where we will study numerically the Ising model  with a spin-decay mechanism provided by $L_n^{(1a)} = \sigma_n^-$ that generalises Eq.  to spins. We restate that this choice of local Lindblad operators does not lead to thermalisation in the steady state, as for this purpose non-local terms would be required. However they cover a special interest since in several experimental implementations, such as circuit-QED, cold-atom settings or trapped ions, this kind of local damping is the relevant one.

In order to quantify the loss of adiabaticity during the annealing protocol, originating both from the closure of the Hamiltonian gap and from the dissipative processes, we are going to study the excess energy $\varepsilon$ per site, at the end of the annealing. The excess energy at a given time $t$ expresses the difference between the instantaneous energy during the annealing, $E(t) = {\rm Tr} \big[ H(t) \, \rho(t) \big]$, where $\rho(t)$ is the solution 