\end{split}$$ As shown in Fig. \[fig:diagnosis\], this final expression can be represented as a graph with the number of vertices equal to the number of logical qubits $n_{l}$ corresponding to the set of variables $\{q_i\}$. In this representation, $Q_{ii}$ can be seen as the weights on the vertices, while $Q_{ij}$ are the weights for the edges representing the couplings between variables $i$ and $j$ (see Fig. \[fig:diagnosis\]). Notice that since $q^2_i = q_i$, the expression $\mathbf{q}^T \cdot \mathbf{Q} \cdot \mathbf{q}$ contains both linear terms $Q_{ii}$, and quadratic term, $Q_{ij}$, when $i \neq j$. $E^0_{\text{QUBO}}$ corresponds to the constant independent term.

A simple mathematical transformation of the form $q_i = (s_i+1)/2$ allows to rewrite $H_{\text{QUBO}}$ in the equivalent problem: $$\label{eq:Hising}
\begin{split}
H_{\text{Ising}}(\{s_i\}) &= E^0_{\text{Ising}} + \sum_{i}  h_{i} s_i + \sum_{i>j}  J_{ij} s_i s_j \\&= E^0_{\text{Ising}} + \mathbf{h} \cdot  \mathbf{s} + \mathbf{s}^T \cdot \mathbf{J} \cdot \mathbf{s}.
\end{split}$$ where one is looking for a minimization over the new set of of so called *spin* variables $\mathbf{s} = \{s_1, s_2, \cdots, s_{n_l}\}$, where now each readout of $s_i = -1 (+1)$ correspond to the assignments $q_i  = 0 (1).$

Results and Discussion {#sec:ResultsDiscussion}
======================

As shown in Fig. \[fig:diagnosis\]c, the quantum device has a 