Abstract
One of the main problems in quantum information systems is the presence of errors due to noise. Many quantum error correcting codes have been designed to deal with generic errors. In this paper we construct new stabilizer codes able to correct a given number e_{mathrm g} of generic Pauli varvec{X}, varvec{Y} and varvec{Z} errors, plus a number e_{mathrm Z} of Pauli errors of a specified type (e.g., varvec{Z} errors). These codes can be of interest when the quantum channel is asymmetric, i.e., when some types of error occur more frequently than others. For example, we design a [[9, 1]] quantum error correcting code able to correct up to one generic qubit error plus one varvec{Z} error in arbitrary positions. According to a generalized version of the quantum Hamming bound, it is the shortest code with this error correction capability.
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