Short Codes for Quantum Channels with One Prevalent Pauli Error Type

TL;DR

This paper introduces a new class of quantum error-correcting codes tailored for channels with asymmetric error types, deriving bounds, proposing design methods, and demonstrating a specific optimal code for correcting mixed error types.

Contribution

It develops a generalized quantum Hamming bound and a syndrome-based design methodology for stabilizer QECCs targeting specific error asymmetries, including a shortest optimal code example.

Findings

Derived a generalized quantum Hamming bound for asymmetric error correction.

Proposed a syndrome assignment-based design methodology for such codes.

Constructed a [[9,1]] code correcting one generic and one Z error, shortest for given parameters.

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 X , Y and Z operators, in this paper we investigate the design of stabilizer QECC able to correct a given number eg of generic Pauli errors, plus eZ Pauli errors of a specified type, e.g., Z 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 [[9,1]] quantum error-correcting code able to correct up to one generic qubit error plus one Z 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.