Asymmetric entanglement-assisted quantum error-correcting codes and BCH codes

TL;DR

This paper introduces asymmetric entanglement-assisted quantum error-correcting codes that leverage error asymmetry and pre-shared entanglement to enhance quantum communication, providing theoretical bounds and explicit BCH code constructions.

Contribution

It presents the concept of asymmetric EAQECCs, derives their parameters, and constructs explicit codes from BCH codes that surpass known bounds.

Findings

Parameters of asymmetric EAQECCs are derived.

Explicit BCH code-based asymmetric EAQECCs are constructed.

Some codes exceed the Gilbert-Varshamov bound.

Abstract

The concept of asymmetric entanglement-assisted quantum error-correcting code (asymmetric EAQECC) is introduced in this article. Codes of this type take advantage of the asymmetry in quantum errors since phase-shift errors are more probable than qudit-flip errors. Moreover, they use pre-shared entanglement between encoder and decoder to simplify the theory of quantum error correction and increase the communication capacity. Thus, asymmetric EAQECCs can be constructed from any pair of classical linear codes over an arbitrary field. Their parameters are described and a Gilbert-Varshamov bound is presented. Explicit parameters of asymmetric EAQECCs from BCH codes are computed and examples exceeding the introduced Gilbert-Varshamov bound are shown.