# Learning and Inference in Hilbert Space with Quantum Graphical Models

arXiv: https://arxiv.org/abs/1810.12369 · HTML: https://mlsanity.com/paper/1810.12369/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2018.

## Built on by (most cited first, 5 of 5)

- [Kernel Density Matrices for Probabilistic Deep Learning](https://mlsanity.com/paper/2305.18204/) (2023)
- [SuperEncoder: Towards Universal Neural Approximate Quantum State Preparation](https://mlsanity.com/paper/2408.05435/) (2024)
- [Robust Iterative Learning Hidden Quantum Markov Models](https://mlsanity.com/paper/2510.23237/) (2025)
- [Nonparametric Topological Layers in Neural Networks](https://mlsanity.com/paper/2111.14829/) (2021)
- [Newton-Schulz Retraction-Based Inference Enables Hidden Quantum Markov Models to Outperform Classical HMMs](https://mlsanity.com/paper/2608.06554/) (2026)

## Abstract (arXiv)

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.