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Quantum graphical models (QGMs) extend the classical framework for reasoningabout uncertainty by incorporating the quantum mechanical view of probability.Prior work on QGMs has focused on hidden quantum Markov models (HQMMs), whichcan be formulated using quantum analogues of the sum rule and Bayes rule usedin classical graphical models. Despite the focus on developing the QGMframework, there has been little progress in learning these models from data.The existing state-of-the-art approach randomly initializes parameters anditeratively finds unitary transformations that increase the likelihood of thedata. While this algorithm demonstrated theoretical strengths of HQMMs overHMMs, it is slow and can only handle a small number of hidden states. In thispaper, we tackle the learning problem by solving a constrained optimizationproblem on the Stiefel manifold using a well-known retraction-based algorithm.We demonstrate that this approach is not only faster and yields bettersolutions on several datasets, but also scales to larger models that wereprohibitively slow to train via the earlier method.

Learning Quantum Graphical Models using Constrained Gradient Descent on the Stiefel Manifold