according to quantum mechanics any measurement of the state collapses the state to a classical outcome (one of the solutions); therefore, if the result is not the desired solution, then one needs to restart another optimization cycle from scratch. This is the main reason why both terms, $H_{\text{consist}}$ and $H_{\text{faults}}$, need to be part of the same cost energy function to be optimized, as expressed in Eq. \[eq:Hproblem\]. Since these quantum algorithms attempt to find the lowest energy solutions, the method we use to cope with this “no-restart" issue is to set the penalty $\lambda_{\text{path}}$ high enough to avoid considering solutions that are inconsistent with the observations. Ideally setting $\lambda_{\text{path}} \rightarrow \infty$ would solve this issue, but due to the limited range of values that can be programmed in the device, this is not possible. At least one should find a lower bound for this penalty such that it guarantees that the desired optimal solution (minimum number of faults satisfying the observation constraints) always has a lower value of $E_{\text{problem}}$ than any assignment violating the observation constraints ($H_{\text{consist}}>0$). For example, for the case of the mapping with uncertainty in the sensors described above in Eq. \[eq:Hconsist\] and Eq. \[eq:Hfaults\], one 