Algebraic geometric construction of a quantum stabilizer code

TL;DR

This paper introduces an algebraic geometric method to construct quantum stabilizer codes using algebraic curves, resulting in improved bounds and asymptotically good code sequences.

Contribution

It presents a novel algebraic geometric construction of self-orthogonal spaces for quantum codes, enhancing code bounds and asymptotic performance.

Findings

Constructed an asymptotically good sequence of binary stabilizer codes.

Improved the Ashikhmin-Litsyn-Tsfasman bound for quantum codes.

Provided a construction method accessible without quantum mechanics knowledge.

Abstract

The stabilizer code is the most general algebraic construction of quantum error-correcting codes proposed so far. A stabilizer code can be constructed from a self-orthogonal subspace of a symplectic space over a finite field. We propose a construction method of such a self-orthogonal space using an algebraic curve. By using the proposed method we construct an asymptotically good sequence of binary stabilizer codes. As a byproduct we improve the Ashikhmin-Litsyn-Tsfasman bound of quantum codes. The main results in this paper can be understood without knowledge of quantum mechanics.