An Algorithmic Approach to Entanglement-Assisted Quantum Error-Correcting Codes from the Hermitian Curve

TL;DR

This paper introduces an efficient algorithm to compute the entanglement parameter for quantum error-correcting codes derived from Hermitian curve-based algebraic geometry codes, enabling the construction of high-quality codes over various fields.

Contribution

It presents a novel algorithmic method to determine the entanglement requirement in Hermitian curve-based EAQECCs, addressing a key unknown parameter in the field.

Findings

The algorithm accurately computes the entanglement parameter c.

It enables the construction of EAQECCs with optimal parameters.

The method applies to codes over any field size.

Abstract

We study entanglement-assisted quantum error-correcting codes (EAQECCs) arising from classical one-point algebraic geometry codes from the Hermitian curve with respect to the Hermitian inner product. Their only unknown parameter is $c$, the number of required maximally entangled quantum states since the Hermitian dual of an AG code is unknown. In this article, we present an efficient algorithmic approach for computing $c$ for this family of EAQECCs. As a result, this algorithm allows us to provide EAQECCs with excellent parameters over any field size.