# On Quantum Circuits for Discrete Graphical Models

arXiv: https://arxiv.org/abs/2206.00398 · HTML: https://mlsanity.com/paper/2206.00398/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 53 references that are in the map.
- First posted on arXiv 1 Jun 2022.

## Builds on (most cited first, 9 of 9)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Quantum generative adversarial networks](https://mlsanity.com/paper/1804.08641/) (2018)
- [Implicit MLE: Backpropagating Through Discrete Exponential Family Distributions](https://mlsanity.com/paper/2106.01798/) (2021)
- [Quantum Inference on Bayesian Networks](https://mlsanity.com/paper/1402.7359/) (2014)
- [Generative training of quantum Boltzmann machines with hidden units](https://mlsanity.com/paper/1905.09902/) (2019)
- [High-quality Thermal Gibbs Sampling with Quantum Annealing Hardware](https://mlsanity.com/paper/2109.01690/) (2021)
- [On Sampling from the Gibbs Distribution with Random Maximum A-Posteriori Perturbations](https://mlsanity.com/paper/1309.7598/) (2013)
- [Quantum algorithm for structure learning of Markov Random Fields](https://mlsanity.com/paper/2109.01014/) (2021)

## Built on by (most cited first, 4 of 4)

- [Expressive equivalence of classical and quantum restricted Boltzmann machines](https://mlsanity.com/paper/2502.17562/) (2025)
- [Computing Marginal and Conditional Divergences between Decomposable Models with Applications](https://mlsanity.com/paper/2310.09129/) (2023)
- [Quantum Boltzmann Machines for Sample-Efficient Reinforcement Learning](https://mlsanity.com/paper/2511.04856/) (2025)
- [An End-to-End Hybrid Quantum-Classical Sampling Workflow for Discrete Markov Random Fields: A Reproducible Case Study](https://mlsanity.com/paper/2607.09893/) (2026)

## Abstract (arXiv)

Graphical models are useful tools for describing structured high-dimensional probability distributions. Development of efficient algorithms for generating unbiased and independent samples from graphical models remains an active research topic. Sampling from graphical models that describe the statistics of discrete variables is a particularly challenging problem, which is intractable in the presence of high dimensions. In this work, we provide the first method that allows one to provably generate unbiased and independent samples from general discrete factor models with a quantum circuit. Our method is compatible with multi-body interactions and its success probability does not depend on the number of variables. To this end, we identify a novel embedding of the graphical model into unitary operators and provide rigorous guarantees on the resulting quantum state. Moreover, we prove a unitary Hammersley-Clifford theorem -- showing that our quantum embedding factorizes over the cliques of the underlying conditional independence structure. Importantly, the quantum embedding allows for maximum likelihood learning as well as maximum a posteriori state approximation via state-of-the-art hybrid quantum-classical methods. Finally, the proposed quantum method can be implemented on current quantum processors. Experiments with quantum simulation as well as actual quantum hardware show that our method can carry out sampling and parameter learning on quantum computers.