Design of Short Codes for Quantum Channels with Asymmetric Pauli Errors

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 [Formula: see text] of generic Pauli [Formula: see text] and [Formula: see text] errors, plus a number [Formula: see text] of Pauli errors of a specified type (e.g., [Formula: see text] 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 [Formula: see text] error in arbitrary positions. According to a generalized version of the quantum Hamming bound, it is the shortest code with this error correction capability.