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