Title: Hybrid Quantum-Classical Stochastic Networks with Boltzmann Layers. Abstract: Quantum Machine Learning (QML) has the potential to significantly advance the state-of-the-art in artificial intelligence, due to recent developments in quantum computing hardware and algorithm design.  Particularly, an avenue opened up by these advances is the possibility of enhancing classical models through developing quantum analogues, which have greater representational power at no extra cost in terms of training and inference.  Here, we investigate analogues of classical networks with stochastic layers, by introducing a class of hybrid stochastic networks that combine layers of several types, including stochastic quantum and classical layers and deterministic classical layers. Further, we introduce Quantum-Annealing (QA)-based sampling techniques that allow such models to be efficiently learned on current QA architectures, using variational and importance-sampling based approaches. Our framework provides benefits in training existing models, including Quantum Boltzmann Machines (QBMs) and Quantum Variational Autoencoders, by allowing local transverse field weights to be optimized jointly with other model parameters, and allows novel hierarchical hybrid models to be learned efficiently. We use classical simulations on synthetic and genomics data to test the impact of including quantum mechanical transverse field terms in such models relative to their classical counterparts. We show that hybrid models are able to achieve better predictive accuracy compared to classical models of matching architecture in these settings, and provide evidence that the local transverse terms can be interpreted as introducing tunable higher-order interactions by connecting genes belonging to common biological pathways.