We develop a quantum version of the em algorithm for training quantum Boltzmann machines. The em algorithm is an information-geometric extension of the well-known expectation-maximization (EM) algorithm, offering a structured alternative to gradient-based methods with potential advantages in stability and convergence. We implement the algorithm on a semi-quantum restricted Boltzmann machine, where quantum effects are confined to the hidden layer. This structure enables analytical update rules while preserving quantum expressivity. Numerical experiments on benchmark datasets show that the proposed method achieves stable learning and outperforms gradient-based training in several cases. These results demonstrate the potential of information-geometric optimization for quantum machine learning, particularly in settings where standard methods struggle due to non-commutativity or vanishing gradients.

翻译：我们提出了一种用于训练量子玻尔兹曼机的量子化EM算法。该EM算法是经典期望最大化（EM）算法在信息几何框架下的扩展，为基于梯度的训练方法提供了一种结构化替代方案，在稳定性和收敛性方面具有潜在优势。我们在一个半量子受限玻尔兹曼机上实现了该算法，其中量子效应被限制在隐藏层。这种结构在保持量子表达力的同时，允许解析更新规则。在基准数据集上的数值实验表明，所提方法实现了稳定学习，并在多种情况下优于基于梯度的训练。这些结果证明了信息几何优化在量子机器学习中的潜力，特别是在非对易性或梯度消失导致标准方法失效的场景中。