π€ AI Summary
This work addresses the challenge of constructing entanglement-assisted quantum error-correcting codes (EAQECCs) for high-dimensional quantum systems in the presence of noisy shared entangled bits (ebits). By generalizing the stabilizer formalism of binary EAQECCs to the qudit case over an arbitrary finite field π½_q, the authors establish a unified construction framework for q-ary EAQECCs that explicitly accounts for ebit noise. Leveraging the generalized Pauli group over π½_q, symplectic geometry, and additive code theory in π½_q^{2n}, this approach systematically generates various families of q-ary EAQECCs. Notably, the resulting codes outperform optimal standard stabilizer codes with equivalent error-correcting capabilities, thereby demonstrating a clear advantage of EAQECCs in specific noisy environments where pre-shared entanglement is imperfect.
π Abstract
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.