New Quantum Codes using Monomial Matrix over Hermitian Dual-Containing Matrix Product Codes

Abstract

Quantum error correction (QEC) is beneficial for ensuring reliable quantum computation and communication by addressing the susceptibility of quantum states to errors from decoherence and noise. This study explores the use of Quantum Matrix Product Codes (MPCs) with an emphasis on monomial matrices and Hermitian duals to achieve efficient and robust error correction. We provide a detailed mathematical formulation of MPCs utilizing monomial matrices, emphasizing the importance of Hermitian duals in maintaining code integrity and ensuring effective error correction. The paper proposes a new criterion for utilizing the rank of the generator matrix associated with linear codes established from Hermitian dual-containing (HDC) of the MPCs. Then, using this criterion, a new set of quantum maximum-distance-separable (MDS) codes has been constructed with better code parameters and error-correction potential. For comparison purpose, different codes have been observed alongside with the obtained codes to ascertain the uniqueness and error correction capabilities.