also restricts the logical noise to be above $10^{-12}$. While this is a very significant improvement, it remains too coarse to be of practical interest. Note moreover that this advantage is much less pronounced when compared to the prediction obtained from infidelity (not shown).

![A function $\epsilon(\cE_0)$ was computed to fit the ansatz of Eq. \[eq:ansatz\] by minimizing the quantity in Eq. \[eq:least\_sq\_fit\] over a training set of $12\times 10^4$ channels, for $\ell=1,2$ and 3 levels of concatenations. Here, we show the correlation $\epsilon(\cE_0)$ to the logical failure rate for $\ell=3$. We see that the ansatz fitted function correlates more tightly with the logical error rate compared to the diamond norm distance, shown for reference.[]{data-label="fig:ansatz"}](Figures/fig5.pdf)

![We have trained a fully connected neural network of 100 nodes and 4 hidden layers with a rectifier (ReLU) [@LBH15], to relate the numerically fitted function $\epsilon(\cE_0)$ shown in Fig. \[fig:ansatz\], to the parameters of the respective the physical CPTP map $\cE_0$ in the training set. To test the efficacy of the trained neural network, we evaluated it on an entirely new ensemble of $6\times 10^4$ channels. Here, we show the logical failure rate as a function of the machine learned function $\epsilon_{\rm predicted}(\cE_0)$ and 