function $\delta^\chi_{ij}(c_i,c_j)$ which assumes unit value only if $(c_i=c_i^\star)$ and $(c_j=c_j^\star)$ and is zero otherwise[^3].

#### Forward Quantum Annealing

The quantum annealing protocol, inspired by the adiabatic principle of quantum mechanics, dictates to drive the system from an initial (easy to prepare) ground state of a quantum Hamiltonian $\mathcal{H}_{\textrm{initial}}$ to the unknown low energy subspace of states of the problem Hamiltonian (\[eq:IsingEmbeddedObjFun\]), ideally to the lowest energy states corresponding to the global minima of the objective function (\[eq:IsingObjFun\]). This *forward* quantum annealing procedure on DW2000Q can be ideally described as attempting to drive the evolution of the following time-dependent Hamiltonian: $$\begin{aligned}
\mathcal{H}_{\textrm{QA}}(t) &=& A[t] \sum_{i=1}^N\left(\sum_{c=1}^{N_c} \bf{\sigma_{ic}^x}\right) + B[t] \mathcal{H}_{\chi\textrm{-Ising}} \; ,
\label{eq:QA}\end{aligned}$$

where the term multiplying $A[t]$ is a Hamiltonian describing an independent collection of local transverse fields for each spin of the system [@johnson2011quantum]. Figure \[fig:schedule\]-a shows how $A[t]$ and $B[t]$ are varying over the scale of total *annealing time* $\tau$ on DW2000Q in units of the typical energy scale of the nominal temperature measured on the chip, 12 milliKelvin. At the end of the annealing run, $A[t]$=0 and the system is projected on the computational basis by a measurement of each qubit magnetization. The duration of the anneal, 