Optimizing short stabilizer codes for asymmetric channels

For a number of quantum channels of interest, phase-flip errors occur far more frequently than bit-flip errors. When transmitting across these asymmetric channels, the decoding error rate can be reduced by tailoring the code used to the channel. However, analyzing the performance of stabilizer codes on these channels is made difficult by the #P-completeness of optimal decoding. To address this, at least for short codes, we demonstrate that the decoding error rate can be approximated by considering only a fraction of the possible errors caused by the channel. Using this approximate error rate calculation, we extend a recent result to show that there are a number of $[[5\leq n\leq12,1\leq k\leq3]]$ cyclic stabilizer codes that perform well on two different asymmetric channels. We also demonstrate that an indication of a stabilizer code's error rate is given by considering the error rate of a classical binary code related to the stabilizer. This classical error rate is far less complex to calculate, and we use it as the basis for a hill climbing algorithm, which we show to be effective at optimizing codes for asymmetric channels. Furthermore, we demonstrate that simple modifications can be made to our hill climbing algorithm to search for codes with desired structure requirements.