# Towards unsupervised representation learning for quantum data: quantum models with inference and generation

arXiv: https://arxiv.org/abs/2609.00372 · HTML: https://mlsanity.com/paper/2609.00372/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 100 references that are in the map.
- First posted on arXiv 31 Aug 2026.

## Builds on (most cited first, 21 of 21)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Self-supervised Visual Feature Learning with Deep Neural Networks: A Survey](https://mlsanity.com/paper/1902.06162/) (2019)
- [A Survey on Contrastive Self-supervised Learning](https://mlsanity.com/paper/2011.00362/) (2020)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [Quantum advantage in learning from experiments](https://mlsanity.com/paper/2112.00778/) (2021)
- [Optimizing the Latent Space of Generative Networks](https://mlsanity.com/paper/1707.05776/) (2017)
- [Experimental Quantum Generative Adversarial Networks for Image Generation](https://mlsanity.com/paper/2010.06201/) (2020)
- [The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine](https://mlsanity.com/paper/1904.02214/) (2019)
- [QNLP in Practice: Running Compositional Models of Meaning on a Quantum Computer](https://mlsanity.com/paper/2102.12846/) (2021)
- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Disintegration and Bayesian Inversion via String Diagrams](https://mlsanity.com/paper/1709.00322/) (2017)
- [Quantum Inference on Bayesian Networks](https://mlsanity.com/paper/1402.7359/) (2014)
- [Variational inference with a quantum computer](https://mlsanity.com/paper/2103.06720/) (2021)
- [Machine learning for protein folding and dynamics](https://mlsanity.com/paper/1911.09811/) (2019)
- [Causal models in string diagrams](https://mlsanity.com/paper/2304.07638/) (2023)
- [A Quantum Extension of Variational Bayes Inference](https://mlsanity.com/paper/1712.04709/) (2017)
- [Quantum Generative Adversarial Autoencoders: Learning latent representations for quantum data generation](https://mlsanity.com/paper/2509.16186/) (2025)
- [Fundamentals of quantum Boltzmann machine learning with visible and hidden units](https://mlsanity.com/paper/2512.19819/) (2025)
- [Discovering quantum phenomena with Interpretable Machine Learning](https://mlsanity.com/paper/2604.16015/) (2026)

## Abstract (arXiv)

With quantum sensors, simulators and networks emerging, a future of quantum technology may produce quantum states as data---that is, coherently rather than as classical measurement records---thus motivating the study of suitable quantum generalisations of modern machine learning, including the automated, unsupervised extraction of useful representations. Two ingredients are central to the latter: inference, mapping observations to latent representations, and generation, mapping latent states back to synthetic data. Both are related to each other and to joint distributions for training models by the chain-rule of classical probability theory. The fact that quantum states however lack such universal, standard factorisation property thus poses a challenge. Here we develop a conceptual and mathematical framework for unsupervised representation learning from quantum data. Models are joint quantum states over visible and latent systems; state-over-time maps provide a notion of factorisation into a marginal state and inference (generation) channel; models with inference (generation) are ambiguous states---states for which such factorisation obtains---subject to a further consistency condition on extended inference maps as data extension. These stipulations are restrictive: we show that non-trivial models must feature non-linear such maps to the extended space. For three representative state-over-time maps, we completely characterise the ambiguous states, uncovering a hierarchy tied to the positive-partial-transpose (PPT) criterion from entanglement theory. Notably, the Leifer-Spekkens construction supports inference and generation exactly for model classes of PPT states, thus allowing genuinely quantum visible-latent correlations. We also formulate quantum counterparts of exact and approximate inference training, explore weaker notions of data extension and sketch a future research programme.