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I have a database of classical codes with simulation results in binary symmetric channel (BSC).

The codes are defined by their parity check matrices $H$. I can pick pairs of codes and call them $H_x$ and $H_z$; if $H_x H_z^T=0$ then these two classical codes define a quantum CSS code.

I already have the simulation results in this form : for every BSC crossover probability $p_{\text{BSC}}$ I have the probability that the decoded codeword is in the classical codespace or not; this is just checking the decoded word syndrome : if $r$=decoded(codeword + BSC noise) then I have the probability of $P_x = P(r H_x^T \neq 0)$ and $P_z=P(r H_z^T \neq 0)$.

So now I have two functions $P_x(p_{BSC})$ and $P_z(p_{BSC})$. I should be able to translate these two curves to the performance of the quantum CSS code but I'm finding it much harder than it should be. A lot of papers are not precise enough about what they mean by "physical failure rate" $p_{fail}$ or "logical error rate" $p_L$; so it's hard to compare against published results. We can pick the depolarizing channel for the quantum code to make things more concrete. If I can translate $p_{BSC}, P_x, P_z$ to $p_{fail}, p_L$ then I can calculate the quantum code threshold by solving for $p_{fail}=p_L$.

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