of DW2000Q, featuring 16x16 unit cells of 8 qubits each, in the ideal case, for a total of 2048 qubits [^2].

To get around this restriction, we employ the minor-embedding compilation technique for fully connected graphs [@boothby2016fast; @venturelli2015spinglass]. By means of this procedure, we obtain another Ising form, where qubits are arranged in ordered 1D chains interlaced on the Chimera graph: $$\begin{aligned}
\mathcal{H}_{\chi\textrm{-Ising}}&=&-\sum_{i=1}^N |J_F| \left[ \sum_{c=1}^{N_c-1} \bf{\sigma_{ic}^z} \sigma_{i(c+1)}^z\right]\label{eq:IsingEmbeddedObjFunJF}\\
&+&\sum_{i=1}^{N} \frac{h_i}{N_c} \left[\sum_{c=1}^{N_c} \bf{\sigma_{ic}^z}\right]
+ \sum_{i,j=1}^{N} J_{ij} \sum_{c_i,c_j=1}^{N_c} \delta_{ij}^\chi(c_i,c_j) \bf{\sigma_{ic_i}^z}\bf{\sigma_{jc_j}^z} \; .
\label{eq:IsingEmbeddedObjFun}\end{aligned}$$

In Eq. (\[eq:IsingEmbeddedObjFunJF\]) we explicitly isolate the encoding of the *logical* quantum variable: the classical binary variable $s_i$ is associated with $N_c$=($N/4+1$) Ising spins $\sigma_{ic}^z$, ferromagnetically coupled directly by strength $J_F$, forming an ordered 1D chain subgraph of $\chi_{16}$. The value of $J_F$ should be strong enough to correlate the value of the magnetization of each individual spin if measured in the computational basis ($\langle\bf{\sigma_{ic}^z}\rangle=\langle\bf{\sigma_{ic^\prime}^z}\rangle$). In the term (\[eq:IsingEmbeddedObjFun\]), we encode the objective function (\[eq:IsingObjFun\]) through our extended set of variables: the local field $h_i$ is evenly distributed across all qubits belonging to the logical chain $i$, and each couplers $J_{ij}$ is active only between one specific pair of qubits $\bf{\sigma_{ic_i^\star}^z}$, $\bf{\sigma_{jc_j^\star}^z}$ which is specified by the adjacency check 