Asymmetric Quantum Concatenated and Tensor Product Codes with Large Z-Distances

TL;DR

This paper introduces a new class of asymmetric quantum codes called AQCTPCs, combining classical concatenated codes with tensor product codes, offering improved error correction, easier construction, and efficient decoding.

Contribution

The paper proposes AQCTPCs, a novel construction that simplifies quantum code design, enhances error correction capabilities, and reduces decoding complexity compared to existing methods.

Findings

AQCTPCs can correct more errors due to high degeneracy.

Decoding complexity is significantly reduced from Ω(n₂a^{n₁}) to O(n₂).

Constructed AQCs outperform known results in literature.

Abstract

In this paper, we present a new construction of asymmetric quantum codes (AQCs) by combining classical concatenated codes (CCs) with tensor product codes (TPCs), called asymmetric quantum concatenated and tensor product codes (AQCTPCs) which have the following three advantages. First, only the outer codes in AQCTPCs need to satisfy the orthogonal constraint in quantum codes, and any classical linear code can be used for the inner, which makes AQCTPCs very easy to construct. Second, most AQCTPCs are highly degenerate, which means they can correct many more errors than their classical TPC counterparts. Consequently, we construct several families of AQCs with better parameters than known results in the literature. Third, AQCTPCs can be efficiently decoded although they are degenerate, provided that the inner and outer codes are efficiently decodable. In particular, we significantly reduce the inner decoding complexity of TPCs from $\Omega(n_2a^{n_1})(a>1)$ to $O(n_2)$ by considering error degeneracy, where $n_1$ and $n_2$ are the block length of the inner code and the outer code, respectively. Furthermore, we generalize our concatenation scheme by using the generalized CCs and TPCs correspondingly.