qecGPT: decoding Quantum Error-correcting Codes with Generative
  Pre-trained Transformers
Hanyan Cao Feng Pan Yijia Wang Pan Zhang
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.