  Quantum error correcting codes are of primary interest for the evolution
towards quantum computing and quantum Internet. We analyze the performance of
stabilizer codes, one of the most important classes for practical
implementations, on both symmetric and asymmetric quantum channels. To this
aim, we first derive the weight enumerator (WE) for the undetectable errors
based on the quantum MacWilliams identities. The WE is then used to evaluate
tight upper bounds on the error rate of CSS quantum codes with minimum weight
decoding. For surface codes we also derive a simple closed form expression of
the bounds over the depolarizing channel. Finally, we introduce a novel
approach that combines the knowledge of WE with a logical operator analysis.
This method allows the derivation of the exact asymptotic performance for short
codes. For example, on a depolarizing channel with physical error rate $\rho
\to 0$ it is found that the logical error rate $\rho_\mathrm{L}$ is
asymptotically $\rho_\mathrm{L} \approx 16 \rho^2$ for the $[[9,1,3]]$ Shor
code, $\rho_\mathrm{L} \approx 16.3 \rho^2$ for the $[[7,1,3]]$ Steane code,
$\rho_\mathrm{L} \approx 18.7 \rho^2$ for the $[[13,1,3]]$ surface code, and
$\rho_\mathrm{L} \approx 149.3 \rho^3$ for the $[[41,1,5]]$ surface code. For
larger codes our bound provides $\rho_\mathrm{L} \approx 1215 \rho^4$ and
$\rho_\mathrm{L} \approx 663 \rho^5$ for the $[[85,1,7]]$ and the
$[[181,1,10]]$ surface codes, respectively.
