Quantifying the Performance of Quantum Codes

We study the properties of error correcting codes for noise models in the presence of asymmetries and/or correlations by means of the entanglement fidelity and the code entropy. First, we consider a dephasing Markovian memory channel and characterize the performance of both a repetition code and an error avoiding code (CRC and CDFS, respectively) in terms of the entanglement fidelity. We also consider the concatenation of such codes (CDFS ◦ CRC) and show that it is especially advantageous in the regime of partial correlations. Finally, we characterize the effectiveness of the codes CDFS, CRC and CDFS ◦ CRC by means of the code entropy and find, in particular, that the effort required for recovering such codes decreases when the error probability decreases and the memory parameter increases. Second, we consider both symmetric and asymmetric depolarizing noisy quantum memory channels and perform quantum error correction via the five qubit stabilizer code C[[5,1,3]]. We characterize this code by means of the entanglement fidelity and the code entropy as function of the asymmetric error probabilities and the degree of memory. Specifically, we uncover that while the asymmetry in the depolarizing errors does not affect the entanglement fidelity of the five qubit code, it becomes a relevant feature when the code entropy is used as a performance quantifier.