# How to translate performance of two classical codes to a quantum CSS code?

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 $$p_{fail}=p_L$$.