Short Codes for Quantum Channels With One Prevalent Pauli Error Type
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Abstract
One of the main problems in quantum information systems is the presence of errors due to noise, and for this reason quantum error-correcting codes (QECCs) play a key role. While most of the known codes are designed for correcting generic errors, i.e., errors represented by arbitrary combinations of Pauli <inline-formula> <tex-math notation="LaTeX">$ {X}, {Y}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$ {Z}$ </tex-math></inline-formula> operators, in this paper we investigate the design of stabilizer QECC able to correct a given number <inline-formula> <tex-math notation="LaTeX">$e_{\mathrm g}$ </tex-math></inline-formula> of generic Pauli errors, plus <inline-formula> <tex-math notation="LaTeX">$e_{\mathrm Z}$ </tex-math></inline-formula> Pauli errors of a specified type, e.g., <inline-formula> <tex-math notation="LaTeX">$ {Z}$ </tex-math></inline-formula> errors. These codes can be of interest when the quantum channel is asymmetric in that some types of error occur more frequently than others. We first derive a generalized quantum Hamming bound for such codes, then propose a design methodology based on syndrome assignments. For example, we found a <inline-formula> <tex-math notation="LaTeX">$[[{9,1}]]$ </tex-math></inline-formula> quantum error-correcting code able to correct up to one generic qubit error plus one <inline-formula> <tex-math notation="LaTeX">$ {Z}$ </tex-math></inline-formula> error in arbitrary positions. This, according to the generalized quantum Hamming bound, is the shortest code with the specified error correction capability. Finally, we evaluate analytically the performance of the new codes over asymmetric channels.