# Quantum Mathematics in Artificial Intelligence

arXiv: https://arxiv.org/abs/2101.04255 · HTML: https://mlsanity.com/paper/2101.04255/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 130 references that are in the map.
- First posted on arXiv 12 Jan 2021.

## Builds on (most cited first, 25 of 25)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Deep contextualized word representations](https://mlsanity.com/paper/1802.05365/) (2018)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Compositionality decomposed: how do neural networks generalise?](https://mlsanity.com/paper/1908.08351/) (2019)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Word Representations via Gaussian Embedding](https://mlsanity.com/paper/1412.6623/) (2014)
- [Foundations for Near-Term Quantum Natural Language Processing](https://mlsanity.com/paper/2012.03755/) (2020)
- [From Word to Sense Embeddings: A Survey on Vector Representations of Meaning](https://mlsanity.com/paper/1805.04032/) (2018)
- [Quantum Natural Language Processing on Near-Term Quantum Computers](https://mlsanity.com/paper/2005.04147/) (2020)
- [Efficient Non-parametric Estimation of Multiple Embeddings per Word in Vector Space](https://mlsanity.com/paper/1504.06654/) (2015)
- [Quantum Deep Learning](https://mlsanity.com/paper/1412.3489/) (2014)
- [RNNs Implicitly Implement Tensor Product Representations](https://mlsanity.com/paper/1812.08718/) (2018)
- [The Frobenius anatomy of word meanings I: subject and object relative pronouns](https://mlsanity.com/paper/1404.5278/) (2014)
- [Domain and Function: A Dual-Space Model of Semantic Relations and Compositions](https://mlsanity.com/paper/1309.4035/) (2013)
- [The Frobenius anatomy of word meanings II: possessive relative pronouns](https://mlsanity.com/paper/1406.4690/) (2014)
- [A Study of Entanglement in a Categorical Framework of Natural Language](https://mlsanity.com/paper/1405.2874/) (2014)
- [Sentence Entailment in Compositional Distributional Semantics](https://mlsanity.com/paper/1512.04419/) (2015)
- [Towards logical negation for compositional distributional semantics](https://mlsanity.com/paper/2005.04929/) (2020)
- [A probabilistic framework for analysing the compositionality of conceptual combinations](https://mlsanity.com/paper/1305.5753/) (2013)
- [Attentive Tensor Product Learning](https://mlsanity.com/paper/1802.07089/) (2018)
- [At the Interface of Algebra and Statistics](https://mlsanity.com/paper/2004.05631/) (2020)
- [Single-Photon Image Classification](https://mlsanity.com/paper/2008.05859/) (2020)
- [Tree Search and Quantum Computation](https://mlsanity.com/paper/1502.01951/) (2015)

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

- [Near-Term Advances in Quantum Natural Language Processing](https://mlsanity.com/paper/2206.02171/) (2022)
- [Quantum Natural Language Processing](https://mlsanity.com/paper/2403.19758/) (2024)
- [Should Semantic Vector Composition be Explicit? Can it be Linear?](https://mlsanity.com/paper/2104.06555/) (2021)
- [Quantum Circuit Components for Cognitive Decision-Making](https://mlsanity.com/paper/2302.03012/) (2023)
- [Grounded learning for compositional vector semantics](https://mlsanity.com/paper/2401.06808/) (2024)

## Abstract (arXiv)

In the decade since 2010, successes in artificial intelligence have been at the forefront of computer science and technology, and vector space models have solidified a position at the forefront of artificial intelligence. At the same time, quantum computers have become much more powerful, and announcements of major advances are frequently in the news. The mathematical techniques underlying both these areas have more in common than is sometimes realized. Vector spaces took a position at the axiomatic heart of quantum mechanics in the 1930s, and this adoption was a key motivation for the derivation of logic and probability from the linear geometry of vector spaces. Quantum interactions between particles are modelled using the tensor product, which is also used to express objects and operations in artificial neural networks. This paper describes some of these common mathematical areas, including examples of how they are used in artificial intelligence (AI), particularly in automated reasoning and natural language processing (NLP). Techniques discussed include vector spaces, scalar products, subspaces and implication, orthogonal projection and negation, dual vectors, density matrices, positive operators, and tensor products. Application areas include information retrieval, categorization and implication, modelling word-senses and disambiguation, inference in knowledge bases, and semantic composition. Some of these approaches can potentially be implemented on quantum hardware. Many of the practical steps in this implementation are in early stages, and some are already realized. Explaining some of the common mathematical tools can help researchers in both AI and quantum computing further exploit these overlaps, recognizing and exploring new directions along the way.