We will also work in units of $\hbar = 1$ and set the energy scale by fixing $J=1$.

It is possible to realise quantum annealing in the Ising model by tuning the parameter $\Gamma(t)$ in time according to a linear ramping, as for example: $$\Gamma(t) = - t / \tau, \quad \text{for} \ \ t \in (-\infty,0] ,
    \label{eq:annealing}$$ where $\tau$ is related to the ramping speed. The choice  ensures that, during the annealing procedure, the system will encounter a critical point in which the ground-state energy gap closes and defects will start to appear. Duality arguments [@Fisher_1995] show that the phase transition occurs at $\Gamma_{\rm c}=1$. The system is driven from a paramagnetic phase, where all the spins are aligned along the field direction $z$ \[i.e., the ground state of the initial Hamiltonian $H(t_{\rm in}) \propto -\sum_n \sigma^z_n$, since $\Gamma(t_{\rm in}) = +\infty$\], to a doubly-degenerate ferromagnetic phase, where the spins are all pointing along the coupling direction $x$ \[i.e., the ground state of the final Hamiltonian $H(t_{\rm fin}) = -\sum_n \sigma^x_n \sigma^x_{n+1}$, since $\Gamma(t_{\rm fin}) = 0$\].

Several papers have already addressed the KZ scaling of defects with $\tau$ in the paradigmatic Ising model, both for the clean 