\alpha t\log_{10}\epsilon(\cE_{0})\right)^{2} \label{eq:least_sq_fit}\end{gathered}$$ over the constants $\log_{10}C_d$, $\alpha$ and the $\log_{10}\epsilon$ of each channel to best fit the data. Figure \[fig:ansatz\] shows the result of this fit for level $\ell=3$. Then, we use one of several machine learning techniques such as *kernel regression*, *k-nearest neighbours* (k-NN) [@A92] and *multi layer perceptron (MLP) regression* [@OM03] to relate $\epsilon(\cE_0)$ to the parameters of $\cE_0$, for all channels in the training set. The trained machine is then used to compute an estimate of $\epsilon(\cE_0)$ for channels in the testing set. Figure \[fig:prediction\] shows the logical noise strength as a function of this machine learned critical parameter, denoted by $\epsilon_{\rm predicted}(\cE_0)$, for noise models in the testing set. Here, the learning was done by a MLP regressor that used a $L2-$regularized square loss function and was implemented using the *scikit-learn* package [@scikit] in Python. The machine learned parameters clearly have a better predictive power than the diamond norm distance, as shown in . For instance, the diamond norm required to achieve a logical noise rate below $10^{-8}$ can sometime yield a logical noise rate as low as $10^{-20}$. In contrast, the condition to achieve a logical noise rate $10^{-8}$ according to the machine-learned parameter 