Performance of CSS Quantum Codes in Asymmetric Channels

In this work, we derive achievability (upper) bounds for the logical error probability (LEP) of finite-length CSS codes for two asymmetric channels. In other words, we prove the existence of [[n, k]] CSS codes whose LEP does not exceed certain bounds that depend on the code and channel parameters. The bounds are derived with the help of finding the average LEPs of CSS codes from different code ensembles.The obtained bounds allow us to optimize code parameters for specific quantum channels. We show that such optimization leads to drastic performance improvements in asymmetric channels. We further show that expurgating bad codes from the ensemble of all CSS codes also leads to large performance improvements in certain channels.