We generalize the stabilizer formalism for entanglement-assisted quantum error-correcting codes with noisy ebits (EAQECCs-Ne) from the binary case to the general $q$-ary case, where $q$ is a prime power. By leveraging the structure of the generalized Pauli group over $\mathbb{F}_q$ and symplectic geometry over $\mathbb{F}_q^{2n}$, we establish a unified framework for constructing EAQECCs-Ne for qudit systems. Equivalent formulations in terms of symplectic geometry over $\mathbb{F}_q$ and additive codes over $\mathbb{F}_q^{2n}$ are derived. We further construct several families of $q$-ary EAQECCs with noise ebits and analyze their performance compared to optimal stabilizer codes. Our results demonstrate that under certain noise conditions, the proposed EAQECCs-Ne can outperform standard stabilizer codes with equivalent error-correcting capability, offering a promising approach for fault-tolerant quantum computation in high-dimensional quantum systems.

翻译：我们将含噪声ebit的纠缠辅助量子纠错码（EAQECCs-Ne）的稳定子形式从二进制情形推广至一般$q$元情形，其中$q$为素数幂。通过利用$\mathbb{F}_q$上广义Pauli群结构及$\mathbb{F}_q^{2n}$上辛几何结构，我们建立了构建qudit系统EAQECCs-Ne的统一框架。推导出基于$\mathbb{F}_q$上辛几何与$\mathbb{F}_q^{2n}$上加性码的等价表述。进一步构建了多类含噪声ebit的$q$元EAQECCs，并分析了它们与最优稳定子码的性能对比。结果表明，在特定噪声条件下，所提出的EAQECCs-Ne能够超越具有同等纠错能力的标准稳定子码，为高维量子系统中的容错量子计算提供了一条有前景的途径。