We propose a general framework for decoding quantum error-correcting codes
with generative modeling. The model utilizes autoregressive neural networks,
specifically Transformers, to learn the joint probability of logical operators
and syndromes. This training is in an unsupervised way, without the need for
labeled training data, and is thus referred to as pre-training. After the
pre-training, the model can efficiently compute the likelihood of logical
operators for any given syndrome, using maximum likelihood decoding. It can
directly generate the most-likely logical operators with computational
complexity $\mathcal O(2k)$ in the number of logical qubits $k$, which is
significantly better than the conventional maximum likelihood decoding
algorithms that require $\mathcal O(4^k)$ computation. Based on the pre-trained
model, we further propose refinement to achieve more accurately the likelihood
of logical operators for a given syndrome by directly sampling the stabilizer
operators. We perform numerical experiments on stabilizer codes with small code
distances, using both depolarizing error models and error models with
correlated noise. The results show that our approach provides significantly
better decoding accuracy than the minimum weight perfect matching and
belief-propagation-based algorithms. Our framework is general and can be
applied to any error model and quantum codes with different topologies such as
surface codes and quantum LDPC codes. Furthermore, it leverages the
parallelization capabilities of GPUs, enabling simultaneous decoding of a large
number of syndromes. Our approach sheds light on the efficient and accurate
decoding of quantum error-correcting codes using generative artificial
intelligence and modern computational power.

我们提出了一种使用生成建模对量子纠错码进行译码的通用框架，该模型利用自回归神经网络，特别是 Transformer，学习逻辑运算符和综合症的联合概率。经过预训练后，该模型可以在给定综合症的情况下，使用最大似然译码高效计算逻辑运算符的可能性，并以计算复杂度为 O (2k)（k 为逻辑比特数）直接生成最可能的逻辑运算符，远优于需要 O (4^k) 计算的传统最大似然译码算法。我们进一步提出了一种基于经过预训练的模型的改进方法，通过直接采样稳定子运算符，更准确地确定给定综合症的逻辑运算符的可能性。我们对具有小码距的稳定子码进行了数值实验，使用了失真性误差模型和具有相关噪声的误差模型。结果显示，我们的方法在译码准确性方面比最小重量完美匹配和基于置信传播的算法提供了显着改进。我们的框架是通用的，可以应用于任何误差模型和具有不同拓扑结构的量子码，如面对码和量子 LDPC 码。此外，它利用了 GPU 的并行化能力，能够同时解码大量的综合症。我们的方法利用了生成式人工智能和现代计算能力对量子纠错码的高效准确解码提供了启示。

qecGPT: 用生成式预训练转换器解码量子纠错码

qecGPT: decoding Quantum Error-correcting Codes with Generative  Pre-trained Transformers

The problem of maximum likelihood decoding with a neural decoder for
error-correcting code is considered. It is shown that the neural decoder can be
improved with two novel loss terms on the node's activations. The first loss
term imposes a sparse constraint on the node's activations. Whereas, the second
loss term tried to mimic the node's activations from a teacher decoder which
has better performance. The proposed method has the same run time complexity
and model size as the neural Belief Propagation decoder, while improving the
decoding performance by up to $1.1dB$ on BCH codes.

研究了神经解码器的最大似然解码问题，通过在节点的激活上使用两个新的损失项，实现了神经解码器的改进；这个改进方法的运行时间和模型大小与 BP 神经解码器相同，但在 BCH 码上将解码性能提高了 1.1dB。

优化节点激活的神经解码

Neural Decoding with Optimization of Node Activations

Linear Programming (LP) is an important decoding technique for binary linear
codes. However, the advantages of LP decoding, such as low error floor and
strong theoretical guarantee, etc., come at the cost of high computational
complexity and poor performance at the low signal-to-noise ratio (SNR) region.
In this letter, we adopt the penalty dual decomposition (PDD) framework and
propose a PDD algorithm to address the fundamental polytope based maximum
likelihood (ML) decoding problem. Furthermore, we propose to integrate machine
learning techniques into the most time-consuming part of the PDD decoding
algorithm, i.e., check polytope projection (CPP). Inspired by the fact that a
multi-layer perception (MLP) can theoretically approximate any nonlinear
mapping function, we present a specially designed neural CPP (NCPP) algorithm
to decrease the decoding latency. Simulation results demonstrate the
effectiveness of the proposed algorithms.

本文提出了基于惩罚对偶分解（PDD）框架的 PDD 算法，并将机器学习技术集成到 PDD 解码算法的最耗时部分 —— 检查多面体投影（CPP）中，提出了特别设计的神经 CPP（NCPP）算法来降低解码延迟。仿真结果表明，所提出的算法是有效的