empirical estimate of $\overline \cN_\ell$.

Standard error metrics {#sec:StandardMetrics}
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Figure \[fig:scatter\] shows the average logical noise strength as a function of the physical noise strength for a wide range of channels and using different measures of noise strength. What we observe is that the logical noise strength varies wildly for a fixed physical noise strength, which implies that estimating the logical noise strength given only the physical noise strength is doomed to yield extremely inaccurate estimates[^2]. We have used several combinations of noise measures – infidelity, diamond norm distance, 1-norm distance, 2-norm distance, entropy, and worst case error – (defined in Sec. \[app:naturalMetrics\]) which all produced similar looking scatter plots. Infidelity was the best metric we found in terms of its ability to predict the behavior of the logical channel, but not by a significant margin.

Focusing on the graphs of c) and d), we reach the conclusion that depolarizing is amongst the worst noise model in the sense that most channels of equal strength result in much less logical noise. This is appealing since the vast majority of numerical simulations to date use the depolarizing channel and furthermore, many of the fault tolerance proofs use the depolarizing channel 