Characterization of $n$-Dimensional Toric and Burst-Error-Correcting Quantum Codes from Lattice Codes
Cibele Cristina Trinca, Reginaldo Palazzo, J. Carmelo Interlando, Ricardo Augusto Watanabe, Clarice Dias de Albuquerque, Edson Donizete de Carvalho, Antonio Aparecido de Andrade
TL;DR
A generalization of a quantum interleaving method for combating clusters of errors in toric quantum error-correcting codes is introduced and can be applied for burst error correction in cases involving localized errors, quantum data storage and quantum channels with memory.
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.