SIAM Journal on Computing, Volume 50, Issue 4, Page 1410-1460, January 2021.
The error threshold of a one-parameter family of quantum channels is defined as the largest noise level such that the quantum capacity of the channel remains positive. This in turn guarantees the existence of a quantum error correction code for noise modeled by that channel. Discretizing the single-qubit errors leads to the important family of Pauli quantum channels; curiously, multipartite entangled states can increase the threshold of these channels beyond the so-called hashing bound, an effect termed superadditivity of coherent information. In this work, we divide the simplex of Pauli channels into one-parameter families and compute numerical lower bounds on their error thresholds. We find substantial increases of error thresholds relative to the hashing bound for large regions in the Pauli simplex corresponding to biased noise, which is a realistic noise model in promising quantum computing architectures. The error thresholds are computed on the family of graph states, a special type of stabilizer state. In order to determine the coherent information of a graph state, we devise an algorithm that exploits the symmetries of the underlying graph, resulting in a substantial computational speed-up. This algorithm uses tools from computational group theory and allows us to consider symmetric graph states on a large number of vertices. Our algorithm works particularly well for repetition codes and concatenated repetition codes (or cat codes), for which our results provide the first comprehensive study of superadditivity for arbitrary Pauli channels. In addition, we identify a novel family of quantum codes based on tree graphs. The error thresholds of these tree graph states outperform repetition and cat codes in large regions of the Pauli simplex, and hence form a new code family with desirable error correction properties.

中文翻译：

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任意泡利噪声的误差阈值

SIAM Journal on Computing，第 50 卷，第 4 期，第 1410-1460 页，2021 年 1 月。
单参数量子信道族的误差阈值被定义为最大噪声水平，使得信道的量子容量保持为正。这反过来又保证了由该通道建模的噪声的量子纠错码的存在。离散单量子位误差导致重要的泡利量子通道家族；奇怪的是，多部分纠缠态可以将这些通道的阈值增加到所谓的散列边界之外，这种效应称为相干信息的超可加性。在这项工作中，我们将泡利通道的单纯形划分为单参数族，并计算其误差阈值的数值下限。我们发现相对于散列边界的误差阈值显着增加，泡利单纯形中对应于偏置噪声的大区域，这是有前途的量子计算架构中的现实噪声模型。误差阈值是在图状态族（一种特殊类型的稳定器状态）上计算的。为了确定图状态的相干信息，我们设计了一种利用底层图的对称性的算法，从而显着提高了计算速度。该算法使用来自计算群论的工具，并允许我们考虑大量顶点上的对称图状态。我们的算法特别适用于重复码和串联重复码（或猫码），我们的结果首次全面研究了任意泡利通道的超可加性。此外，我们基于树图确定了一个新的量子代码族。