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.

翻译：我们提出了一种利用生成式建模解码量子纠错码的通用框架。该模型采用自回归神经网络（特别是Transformer）以无监督方式学习逻辑算子和综合征的联合概率，无需标注训练数据，故称为预训练。预训练后，模型可高效计算任意给定综合征下逻辑算子的似然，并实现最大似然解码。它可直接生成最可能的逻辑算子，计算复杂度为$\mathcal O(2k)$（其中$k$为逻辑量子比特数），显著优于传统最大似然解码算法所需的$\mathcal O(4^k)$计算量。基于预训练模型，我们进一步提出通过直接采样稳定子算子的细化方法，更精确地获得给定综合征下逻辑算子的似然。我们针对小码距稳定子码进行了数值实验，分别采用去极化错误模型和含相关噪声的错误模型。结果表明，我们的方法在解码精度上显著优于最小权重完美匹配和基于置信传播的算法。该框架具有通用性，可适用于任意错误模型及不同拓扑结构的量子码（如表面码和量子LDPC码）。此外，它利用GPU的并行计算能力，可同时解码大量综合征。我们的方法为利用生成式人工智能与现代计算能力实现高效准确的量子纠错码解码提供了新思路。