times instead of evaluating the sum directly, which provides an estimate $\widetilde \cE_{\ell} = \frac{1}{N} \sum_j \cE^{s_j}_{\ell}$ of $\overline \cE_{\ell}$, where $s_j$ denotes the sampled syndromes.

Critical noise parameters {#sec:criticalparameters}
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A very coarse description of a noise model $\cE$ would be a single number specifying its “noise level", or “strength”, with strength 0 corresponding to a noiseless channel (the identity map). There are several inequivalent measures which are used to describe the strength. Some, such as the average infidelity, are efficiently accessible experimentally [@EAZ05; @KLR08; @WT14]. Others, like the diamond distance [@Kit97; @W09; @G05], are more convenient mathematically but much more challenging to probe experimentally. Let us denote by $\cN$ such generic noise measure, i.e., $\cN(\cE)$ is the noise strength of the CPTP map $\cE$.

Heuristically, the fault-tolerant accuracy threshold theorem [@AB08; @KLZ96; @K97; @Pres98] states that, provided the noise strength $\cN_0 \equiv \cN(\cE_0)$ of the physical channel is less than a certain threshold value, the average logical noise strength $\overline \cN_\ell$ will decrease doubly exponentially with the level of concatenation $\ell$. This theorem is proved either assuming a stochastic noise model [@AGP07] – in which case all metrics are essentially equivalent – or using the diamond norm distance 