An Analysis of Shor and Steane Code Performance under Coherent Errors

We derive explicit process matrices (effective channels) for the Shor code (a nine-qubit code) and the Steane code (a seven-qubit code) under any unital error channel applied to each physical qubit. These matrices provide valuable insights into the performance of these codes. The derived process matrix enables a rigorous proof that a concatenated code with a symmetric decoder can map an open set of arbitrary error channels to the identity channel. This result generalizes a previously established theorem, which was limited to an open set of diagonal error channels. For commonly studied coherent error models, we leverage the process matrices to perform precise analyses of code performance in terms of average gate infidelity and diamond distance, comparing the physical error channels to the resulting effective channels after error correction. These results refine and extend related findings in prior works.