# Entanglement-assisted Quantum Codes from Algebraic Geometry Codes

## TL;DR

This paper constructs new entanglement-assisted quantum error-correcting codes using algebraic geometry codes, achieving high rates and asymptotic goodness that surpass known bounds like the quantum Gilbert-Varshamov bound.

## Contribution

It introduces novel families of QUENTA codes from algebraic geometry codes with maximal entanglement and optimal parameters, surpassing existing bounds.

## Key findings

- Some codes have quantum Singleton defect zero or one.
- Codes surpass the quantum Gilbert-Varshamov bound in rate.
- Asymptotically good families of codes are constructed.

## Abstract

Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum (QUENTA) codes. In this paper, we use algebraic geometry codes to construct several families of QUENTA codes via the Euclidean and the Hermitian construction. Two of the families created have maximal entanglement and have quantum Singleton defect equal to zero or one. Comparing the other families with the codes with the respective quantum Gilbert-Varshamov bound, we show that our codes have a rate that surpasses that bound. At the end, asymptotically good towers of linear complementary dual codes are used to obtain asymptotically good families of maximal entanglement QUENTA codes. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that using our construction it is possible to create an asymptotically family of QUENTA codes that exceeds this bound.