Entanglement-Assisted Concatenated Quantum Codes: Parameters and Asymptotic Performance

TL;DR

This paper introduces new entanglement-assisted concatenated quantum codes (EACQCs) with improved parameters and asymptotic performance, constructed using almost MDS and $ ext{h}$-MDS codes, and demonstrates their optimality and asymptotic bounds.

Contribution

It presents novel EACQCs with better parameters than existing codes, including maximal-entanglement and asymptotically good families, using advanced algebraic geometry codes.

Findings

Constructed EACQCs outperform previous quantum codes in parameters.

Developed maximal-entanglement EACQCs with optimal or near-optimal minimum distances.

Proved EACQCs can asymptotically attain the quantum Gilbert-Varshamov bound.

Abstract

Entanglement-assisted concatenated quantum codes (EACQCs) are constructed by concatenating two entanglement-assisted quantum error-correcting codes (EAQECCs). By selecting the inner and outer component codes carefully, it is able to construct state-of-the-art EACQCs with parameters better than previous quantum codes. In this work, we use almost maximum-distance-separable (MDS) codes and $\hbar$-MDS codes as the outer codes to construct EACQCs. Because the range of code length of almost MDS and $\hbar$-MDS codes is much more free than that of the commonly used MDS codes. We derive several families of new EACQCs with parameters better than the previously best known EAQECCs and standard quantum error-correcting codes (QECCs) of the same length and net transmissions. Moreover, we demonstrate that EACQCs are with maximal entanglement if both the inner and outer component codes are with maximal entanglement. As a result, we construct three new maximal-entanglement EACQCs which have optimal parameters. In addition, we present several new maximal-entanglement EACQCs whose minimum distance is only one less than the minimum distance of the optimal codes. In particular, we propose two new families of asymptotically good maximal-entanglement EACQCs with explicit constructions by using entanglement-assisted quantum algebraic geometry codes as the outer codes. At last, we prove that EACQCs can attain the quantum Gilbert-Varshamov bound for EAQECCs asymptotically.