up to a point where the Hamiltonian time-dependence is temporarily paused. With reference to the Hamiltonian evolution in Eq.(\[eq:QA\]), the transverse field evolution that we program[^4] for this protocol is the following three-phases function (identical equations for $B^R[t]$, $B^p[t]$, $B^F[t]$ in terms of $B[t]$): $$\begin{aligned}
A^R[t]=A[\tau+(s_p-1)t]&\;\;&\textrm{(Reverse Annealing)}\;\; t\in[0,\tau]\label{eq:reverseSchedule}\\
A^p[t]=A[s_p\tau]&\;\;&\textrm{(Annealing Pause)}\;\; t\in[\tau,\tau+\rho]\nonumber\\
A^F[t]=A[(1-s_p)(t-\rho)-(1-2s_p)\tau]&\;\;&\textrm{(Forward Annealing)}\;\; 
t\in[\tau+\rho,2\tau+\rho]\nonumber\end{aligned}$$ where $\rho$ is the duration of the pause and $s_p\in[0,1]$ indicates the location of the forward schedule where the pause is implemented. The total duration of the selected reverse anneal protocol is $2\tau+\rho$ as opposed to $\tau$ for the forward anneal.

While the theory of reverse annealing is just starting to be investigated [@marshall2018pausing; @ohkuwa2018reverse; @kechedzhi2018efficient], the physics rationale of reverse annealing is to be found in the oversimplified idea that, if the system is initialized in a state $\mathcal{S}$ corresponding to a local minimum of the objective function (\[eq:IsingObjFun\]), the interplay of quantum and thermal fluctuations might help the state *tunnel* out of the energy trap during the reverse annealing, while the annealing pause (and to some extent the final forward annealing) allows the system to thermalize and relax in the neighborhood of the newfound minimum. See Figure [\[fig:schedule\]-c]{} for a pictorial view of the 