These $6$ total failures can be any combination of sensors equal to $0$ or outermost CB’s equal to $0$, as long as the total number summed amounts to $6$ and each failure corresponds to exactly one of the branches with $l_i=0$.

Notice the pseudo-boolean $H_{\text{consist}}$ is a high-degree polynomial, and for this particular network, the order of the polynomial is related to the depth of the tree. We can reduce the degree of the polynomial to a quadratic expression, $H_{\text{QUBO}}$, with the overhead of adding more binary variables, while conserving the global minimum of the original pseudo-boolean function, $H(\{x_i\},\{y_i\};\{l_i\})$. Further details on the new techniques used for this reduction are provided in the appendix.

Assuming it requires $n_A$ ancilla variables $\{a_i\}$ to reduce the high-degree polynomial to the quadratic expression, we can relabel the CB, sensor, and ancilla variables, $\{x_i\}$, $\{y_i\}$, and $\{a_i\}$, respectively, into a new set of binary variables $\{q_i\}$ for $i = 1,2,3, \cdots, n_{l}$, with $n_{l} = n_{\text{CB}}+ n_{\text{sensor}}+n_{A}$ as the total number of logical qubits. The final quadratic cost function to be minimized can be written as $$\label{eq:Hqubo}
\begin{split}
H_{\text{QUBO}}(\{q_i\}) &= E^0_{\text{QUBO}} + \sum_{i,j}  Q_{ij} q_i q_j \\&= E^0_{\text{QUBO}} + \mathbf{q}^T \cdot \mathbf{Q} \cdot \mathbf{q}.
