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The pivotal challenge of handling uncertainty in artificial intelligence has prompted a growing focus on learning Bayesian network (BN) structures from data in recent years. Nevertheless, most of the existing methods such as hill-climbing, variational auto-encoder and graph neural network-based algorithms continue to encounter the issues like local optimum and exponential computational complexity, etc. In response to these challenges, we propose a Transformer-based approach for learning the BN structure (T-BNSL) from an information theory perspective. Specifically, we first establish the graph skeleton of a BN by employing conditional independence tests grounded in mutual information. Subsequently, we propose a hill-climbing method with decomposable BIC scoring function to generate potential directed acyclic graphs (DAGs) and evaluate the BIC scores of a subset of these DAGs. Lastly, we use the Transformer framework with a refined attention module to predict the BIC scores of the remaining DAGs, enabling the efficient identification of the DAG with the highest score. Experimental findings demonstrate that our approach significantly surpasses state-of-the-art competitors in terms of accuracy and efficiency by several orders of magnitude when learning the BN structure.