a function of $\cE_0$, but it is very difficult to compute even in an oversimplified model as explained in (see also ).

Motivated by the fault-tolerance accuracy threshold theorem, we make the following ansatz for the behaviour of the logical noise. For a physical noise model $\cE_0$ and given a fault-tolerant protocol family with increasing minimal distance $d$ (for the concatenated Steane code, $d = 3^\ell$), the logical noise strength decreases as $$\overline \cN_d = C_\ell [\epsilon(\cE_0)]^{\alpha t} \label{eq:ansatz}$$ where $t = \lfloor(d-1)/2\rfloor + 1, C_d$ and $\alpha$ are positive constants which are specific to the fault-tolerant scheme, while $\epsilon(\cE_0)$ is a critical parameter of the physical noise model. Our goal is thus to find this function $\epsilon$. This proceeds in two steps. We consider two arbitrary set of randomly generated physical channels, one called the *training set* which in our case is the same as the one studied in . The other is called the *testing set* which in our case is a different ensemble that is half the size of the training set. On the training set, we perform a least square fit of the ansatz in Eq. \[eq:ansatz\] which minimizes the function $$\begin{gathered}
\sum_{\cE_0,d}\left(\log_{10}\widetilde{\cN}_{\ell} - \log_{10}C_{d} - 