Learning and Inference in Hilbert Space with Quantum Graphical Models
This work addresses the challenge of probabilistic modeling in quantum systems for researchers in quantum machine learning, though it appears incremental as it builds on existing quantum graphical models and Hilbert space embeddings.
The paper tackles the problem of modeling uncertainty and dynamics in quantum graphical models by linking them to Hilbert space embeddings, showing that quantum inference rules correspond to kernel operations. It proposes a nonparametric method, HSE-HQMM, which achieves competitive performance with state-of-the-art models like LSTMs and PSRNNs on several datasets.
Quantum Graphical Models (QGMs) generalize classical graphical models by adopting the formalism for reasoning about uncertainty from quantum mechanics. Unlike classical graphical models, QGMs represent uncertainty with density matrices in complex Hilbert spaces. Hilbert space embeddings (HSEs) also generalize Bayesian inference in Hilbert spaces. We investigate the link between QGMs and HSEs and show that the sum rule and Bayes rule for QGMs are equivalent to the kernel sum rule in HSEs and a special case of Nadaraya-Watson kernel regression, respectively. We show that these operations can be kernelized, and use these insights to propose a Hilbert Space Embedding of Hidden Quantum Markov Models (HSE-HQMM) to model dynamics. We present experimental results showing that HSE-HQMMs are competitive with state-of-the-art models like LSTMs and PSRNNs on several datasets, while also providing a nonparametric method for maintaining a probability distribution over continuous-valued features.