[@AB08; @SDT07; @AP09] in place of $\cN$. The theorem makes conservative assumptions about the nature of the noise model, and it at best provides very loose upper bounds on $\cN(\overline \cE_\ell)$. Upper bounds are of very little use in order to optimize a fault tolerant scheme, so we would like to develop a better understanding of the behaviour of $\overline \cN_\ell$.

In the previous section, we described a numerical procedure to sample logical channels $\cE^s_\ell$ corresponding to $\ell$ levels of concatenation. This gives us a mean to estimate, within statistical errors, the average logical noise strength. Note that there are two natural definitions of the average logical noise strength, either as the noise strength of the average channel [@CGFF17] $$\cN(\overline \cE_\ell) = \cN\left(\sum_s \Pr(s) \cE_\ell^s\right),
\label{eq:N1}$$ or as the average of the noise strength over the different syndromes [@GSCL16] $$\overline{\cN( \cE_\ell)} = \sum_s \Pr(s) \cN(\cE_\ell^s).
\label{eq:N2}$$ We have used both definitions in our numerical simulations, and this choice has quantitative but no qualitative effect on our conclusions. The results presented in the rest of this article use the measure of , but we will continue to use the generic notation $\overline \cN_\ell$. As above, we will denote $\widetilde \cN_\ell$ the 