Nonbinary Quantum Cyclic and Subsystem Codes Over Asymmetrically-decohered Quantum Channels

TL;DR

This paper develops methods to construct asymmetric quantum codes, including cyclic, BCH, RS, and RM codes, to protect quantum information over channels with asymmetric noise, enhancing quantum error correction.

Contribution

It introduces two generic methods for constructing asymmetric quantum cyclic codes from classical cyclic codes, enabling the creation of various quantum BCH, RS, and RM codes.

Findings

Constructed multiple families of asymmetric quantum cyclic codes.

Developed methods for deriving asymmetric quantum BCH, RS, and RM codes.

Constructed families of asymmetric subsystem codes.

Abstract

Quantum computers theoretically are able to solve certain problems more quickly than any deterministic or probabilistic computers. A quantum computer exploits the rules of quantum mechanics to speed up computations. However, one has to mitigate the resulting noise and decoherence effects to avoid computational errors in order to successfully build quantum computers. In this paper, we construct asymmetric quantum codes to protect quantum information over asymmetric quantum channels, $\Pr Z \geq \Pr X$. Two generic methods are presented to derive asymmetric quantum cyclic codes using the generator polynomials and defining sets of classical cyclic codes. Consequently, the methods allow us to construct several families of quantum BCH, RS, and RM codes over asymmetric quantum channels. Finally, the methods are used to construct families of asymmetric subsystem codes.