  One of the main challenge for an efficient implementation of quantum
information technologies is how to counteract quantum noise. Quantum error
correcting codes are therefore of primary interest for the evolution towards
quantum computing and quantum Internet. We here analyze the performance of
surface codes, one of the most important class for practical implementations,
on both symmetric and asymmetric quantum channels. We derive approximate
expressions, confirmed by simulations, to evaluate the performance of surface
codes and of XZZX codes, and provide a metric to assess the advantage of codes
with respect to uncoded systems. Our findings allow to characterize the
performance by means of analytical formulas of surface codes, like, for
example, the [[13, 1, 3]], the [[23, 1, 3/5]], the [[33, 1, 3/7]], and the
[[41, 1, 5]] surface codes.
