Characterization of $n$-Dimensional Toric and Burst-Error-Correcting Quantum Codes from Lattice Codes

TL;DR

This paper introduces new $n$-dimensional toric quantum codes derived from lattice codes, applying a generalized quantum interleaving method to enhance burst-error correction, with implications for quantum data storage and channels with memory.

Contribution

The work generalizes quantum interleaving to $n$-dimensional toric codes and derives novel burst-error-correcting codes with high code rate and gain.

Findings

Developed $n$-dimensional toric quantum codes from lattice codes.

Applied quantum interleaving to create burst-error-correcting codes.

Potential uniqueness of these codes if the Golomb-Welch conjecture holds.

Abstract

Quantum error correction is essential for the development of any scalable quantum computer. In this work we introduce a generalization of a quantum interleaving method for combating clusters of errors in toric quantum error-correcting codes. We present new $n$-dimensional toric quantum codes, where $n\geq 5$, which are featured by lattice codes and apply the proposed quantum interleaving method to such new $n$-dimensional toric quantum codes. Through the application of this method to these novel $n$-dimensional toric quantum codes we derive new $n$-dimensional quantum burst-error-correcting codes. Consequently, $n$-dimensional toric quantum codes and burst-error-correcting quantum codes are provided offering both a good code rate and a significant coding gain when it comes to toric quantum codes. Another important consequence from the presented $n$-dimensional toric quantum codes is that if the Golomb and Welch conjecture in \cite{perfcodes} regarding the Lee sphere in $n$ dimensions for the respective close packings holds true, then it follows that these $n$-dimensional toric quantum codes are the only possible ones to be obtained from lattice codes. Moreover, such a methodology can be applied for burst error correction in cases involving localized errors, quantum data storage and quantum channels with memory.