Asymmetric Quantum Cyclic Codes

TL;DR

This paper introduces methods for constructing asymmetric quantum cyclic codes tailored for channels where phase-shift errors are more probable than qubit-flip errors, enhancing quantum error correction.

Contribution

It presents two generic methods to derive asymmetric quantum cyclic codes from classical cyclic codes, enabling the construction of various families of asymmetric quantum BCH, RS, RM, and subsystem codes.

Findings

Constructed several families of asymmetric quantum BCH, RS, and RM codes.

Developed methods for deriving asymmetric quantum codes from classical cyclic codes.

Enabled the creation of asymmetric subsystem codes.

Abstract

It is recently conjectured in quantum information processing that phase-shift errors occur with high probability than qubit-flip errors, hence the former is more disturbing to quantum information than the later one. This leads us to construct asymmetric quantum error controlling codes to protect quantum information over asymmetric channels, $\Pr Z \geq \Pr X$. In this paper we present two generic methods 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 asymmetric quantum BCH, RS, and RM codes. Finally, the methods are used to construct families of asymmetric subsystem codes.