Perfect $\mathcal{E}$-Error-Correcting Quantum Codes over the Hurwitz Integers

Abstract

The concept of perfect $\mathcal{E}$-error-correcting codes constitutes a fundamental benchmark in classical coding theory; however, a corresponding definition for these codes remains absent in the quantum domain. This study addresses this theoretical gap by formally defining perfect $\mathcal{E}$-error-correcting quantum codes and constructing a new family of codes that satisfy this definition. We derive these quantum codes from classical linear codes generated over residue class rings of prime Hurwitz integers. The newly constructed quantum codes are presented with their parameters, and those qualifying as perfect $\mathcal{E}$-error-correcting codes are identified. Additionally, this study designs quantum logic gates tailored for the proposed codes over Hurwitz integers. These findings extend the algebraic utility of Hurwitz integers in quantum information theory and provide a novel class of quantum codes equipped with their fundamental logical operators.