Nonbinary stabilizer codes over finite fields

TL;DR

This paper develops the theory of nonbinary stabilizer quantum codes over finite fields, providing new characterizations, bounds, and constructions that extend the binary case to more general settings.

Contribution

It introduces a Galois-theoretic framework, generalizes additive codes over GF(4), and constructs various families of nonbinary stabilizer codes with bounds and puncturing theory.

Findings

Characterization of nonbinary stabilizer codes over GF(q)

Construction of quantum Hamming, quadratic residue, and BCH codes

Bounds on minimum distance and maximal length of MDS stabilizer codes

Abstract

One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over GF(q) in terms of classical codes over GF(q^2) is provided that generalizes the well-known notion of additive codes over GF(4) of the binary case. This paper derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum BCH codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.