of the master equation  at time $t$, and the ground-state energy $E_0(t) = {\rm Tr} \big[ H(t) |\psi_0(t)\rangle \langle \psi_0(t)| \big]$ of the instantaneous Hamiltonian system described by $H(t)$: $$\label{eq:excess_energy}
  \varepsilon(t) = \tfrac{1}{L} \Big\{ {\rm Tr} [ H(t) \, \rho(t) ]
  - \langle \psi_0(t)|H(t)|\psi_0(t) \rangle \Big\}.$$ Using the aforementioned Bogoliubov transformation, the second term of Eq.  can be computed straightforwardly, while the first term $E(t)$ is non trivial (see the Appendix). For the Ising spin system we will resort to a fully numerical MPO approach. We point out that a related quantity of interest is the density of defects $\mathcal{N} \equiv \tfrac{1}{2L} \sum_{n} \langle 1-\sigma^{x}_{n}\sigma^{x}_{n+1} \rangle$, which, in the case of ordered chains and at the end of the annealing, is equivalent to the excess energy $\varepsilon(0)$, apart from trivial constants.

Free-fermionic system {#sec:TraslInv}
=====================

We first analyse a fermionic system described by the Hamiltonian in Eq. , where anti-periodic boundary conditions are imposed. A Fourier transform drastically helps in the diagonalisation of the unitary problem, since the different momentum modes decouple (see App. \[app:TraslInv\_U\]). Note that for the sake of simplicity, here we only consider one-dimensional systems, but our analysis of fermions can be easily extended to larger dimensionalities, 