# From Conceptual Spaces to Quantum Concepts: Formalising and Learning Structured Conceptual Models

arXiv: https://arxiv.org/abs/2401.08585 · HTML: https://mlsanity.com/paper/2401.08585/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 69 references that are in the map.
- First posted on arXiv 6 Nov 2023.

## Builds on (most cited first, 17 of 17)

- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [QNLP in Practice: Running Compositional Models of Meaning on a Quantum Computer](https://mlsanity.com/paper/2102.12846/) (2021)
- [Imitating Interactive Intelligence](https://mlsanity.com/paper/2012.05672/) (2020)
- [Quantum Variational Autoencoder](https://mlsanity.com/paper/1802.05779/) (2018)
- [Category Theory in Machine Learning](https://mlsanity.com/paper/2106.07032/) (2021)
- [Disintegration and Bayesian Inversion via String Diagrams](https://mlsanity.com/paper/1709.00322/) (2017)
- [Interacting Conceptual Spaces I : Grammatical Composition of Concepts](https://mlsanity.com/paper/1703.08314/) (2017)
- [Resolving References to Objects in Photographs using the Words-As-Classifiers Model](https://mlsanity.com/paper/1510.02125/) (2015)
- [Embedding Words as Distributions with a Bayesian Skip-gram Model](https://mlsanity.com/paper/1711.11027/) (2017)
- [Conversational Negation using Worldly Context in Compositional Distributional Semantics](https://mlsanity.com/paper/2105.05748/) (2021)
- [Composing Conversational Negation](https://mlsanity.com/paper/2107.06820/) (2021)
- [A Thorough Formalization of Conceptual Spaces](https://mlsanity.com/paper/1706.06366/) (2017)
- [A Categorical Semantics of Fuzzy Concepts in Conceptual Spaces](https://mlsanity.com/paper/2110.05985/) (2021)
- [An Introduction to Effectus Theory](https://mlsanity.com/paper/1512.05813/) (2015)
- [Categorical Operational Physics](https://mlsanity.com/paper/1902.00343/) (2019)

## Built on by (most cited first, 3 of 3)

- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [From Mechanistic to Compositional Interpretability](https://mlsanity.com/paper/2605.08934/) (2026)
- [XXAI: Towards eXplicitly eXplainable Artificial Intelligence](https://mlsanity.com/paper/2401.03093/) (2024)

## Abstract (arXiv)

In this article we present a new modelling framework for structured concepts using a category-theoretic generalisation of conceptual spaces, and show how the conceptual representations can be learned automatically from data, using two very different instantiations: one classical and one quantum. A contribution of the work is a thorough category-theoretic formalisation of our framework. We claim that the use of category theory, and in particular the use of string diagrams to describe quantum processes, helps elucidate some of the most important features of our approach. We build upon Gardenfors' classical framework of conceptual spaces, in which cognition is modelled geometrically through the use of convex spaces, which in turn factorise in terms of simpler spaces called domains. We show how concepts from the domains of shape, colour, size and position can be learned from images of simple shapes, where concepts are represented as Gaussians in the classical implementation, and quantum effects in the quantum one. In the classical case we develop a new model which is inspired by the Beta-VAE model of concepts, but is designed to be more closely connected with language, so that the names of concepts form part of the graphical model. In the quantum case, concepts are learned by a hybrid classical-quantum network trained to perform concept classification, where the classical image processing is carried out by a convolutional neural network and the quantum representations are produced by a parameterised quantum circuit. Finally, we consider the question of whether our quantum models of concepts can be considered conceptual spaces in the Gardenfors sense.