Entanglement-assisted quantum error-correcting codes from units

TL;DR

This paper presents a method to construct entanglement-assisted quantum error-correcting codes with customizable rate, error correction, and entanglement, including MDS codes over prime fields, with asymptotic properties.

Contribution

It introduces a systematic construction of EAQECCs with specified parameters and demonstrates their asymptotic behavior and construction over prime fields.

Findings

Constructed EAQECCs with desired rate and error correction

Codes approach non-zero rate and relative distance asymptotically

Codes can be built over prime fields using modular arithmetic

Abstract

Entanglement-assisted quantum error-correcting codes (EAQECCs) to desired rate, error-correcting capability and maximum shared entanglement are constructed. Thus for a required rate $R$, required error-correcting capability to correct $t$ errors, mds (maximum distance separable) EAQECCs of the form $[[n,r,d;c]]$ with $R=\frac{r}{n}, d\geq (2t+1), c = (n-r), d= (n-r+1)$ are constructed. Series of such codes may be constructed where the rate and the relative distance approach non-zero constants as $n$ approaches infinity. The codes may also be constructed over prime order fields in which modular arithmetic may be employed.