Here, I want to ask a basic question about how to compute the fault tolerance threshold of quantum codes. As I know, maybe, the most usual way is to do some simulations. Howvever, I am more intersted with the theoretical computation. More specificly, I am interested with the quantum LDPC relted codes, e.g., https://arxiv.org/abs/1412.6172, https://arxiv.org/abs/1711.08351, and https://arxiv.org/abs/1310.2984. For example, let us talk about
Efficient decoding of random errors for quantum expander codes, by Fawzi,Grospellier,and Leverrier, STOC’18.
My first question is about the failure probability $\epsilon$ (or success probability $1-\epsilon$ in verse). I am confused by the computation of that. Do we need to count all the random errors that can not be corrected succesfully? In that paper, the authors can correct random errors of linear weight with a very high probability $1-\epsilon$. Denote by $\delta n$ the weight of the random errors, where $0<\delta\leq1$ and $n$ is the block length. In general, we can assume that $0<\delta<0.5$ (Is it right or necessary?). Then do we need to deal with the random errors of weight larger than $\delta n$ up to $n$?