Exact performance of the five-qubit code with coherent errors

We derive explicit process matrices (effective logical channels) for five-qubit code under any unital error channel imposed on each physical qubit of the code, which would enable us to gain a lot of insight about the performance of the code. To our best knowledge, this is the first explicit effective logical channel that has been derived for a quantum correction code under a broad class of noise models. The process matrix allows us to rigorously prove that the concatenated code with a symmetric decoder can take an open set of any type error channels to the identity channel. This result extends the theorem that is confined to an open set of diagonal error channels proved by previous authors. For some commonly considered coherent error models, using the process matrices, we conduct precise analysis of the code's performances in terms of average gate infidelity and diamond distance measures for both the physical error channels and the resulting effective logical channels after error correction. These outcomes sharpen the related results shown in recent publications. Finally, we show that an optimal (non-symmetric) decoder of the code achieves substantially improved performance of error correction against a specified noise model, but it no longer corrects an open set of general errors. The results confirm some findings of previous authors who arrived at similar conclusions through different approaches.