This paper presents a mathematical formalism for generative artificial intelligence (GAI). Our starting point is an observation that a “histories” approach to physical systems agrees with the compositional nature of deep neural networks. Mathematically, we define a GAI system as a family of sequential joint probabilities associated with input texts and temporal sequences of tokens (as physical event histories). From a physical perspective on modern chips, we then construct physical models realizing GAI systems as open quantum systems. Finally, as an illustration, we construct physical models realizing large language models based on a transformer architecture as open quantum systems in the Fock space over the Hilbert space of tokens. Our physical models underlie the transformer architecture for large language models.
This paper presents a mathematical formalism for generative artificial intelligence (GAI). Our starting point is an observation that a “histories” approach to physical systems agrees with the compositional nature of deep neural networks. Mathematically, we define a GAI system as a family of sequential joint probabilities associated with input texts and temporal sequences of tokens (as physical event histories). From a physical perspective on modern chips, we then construct physical models realizing GAI systems as open quantum systems. Finally, as an illustration, we construct physical models realizing large language models based on a transformer architecture as open quantum systems in the Fock space over the Hilbert space of tokens. Our physical models underlie the transformer architecture for large language models.
The introduction of the transformer architecture in 2017 marked the most striking advancement in natural language processing. The transformer is a model architecture relying entirely on an attention mechanism to draw global dependencies between input and output. However, we believe there is a gap in our theoretical understanding of what the transformer is, and how it works physically. From a physical perspective on modern chips, such as those chips under 28nm, modern intelligent machines should be regarded as open quantum systems beyond conventional statistical systems. Thereby, in this paper, we construct physical models realizing large language models based on a transformer architecture as open quantum systems in the Fock space over the Hilbert space of tokens. Our physical models underlie the transformer architecture for large language models.
We show that deep neural networks, including transformers and RNNs, pretrained as usual on next-token prediction, intrinsically discover and represent beliefs over 'quantum' and 'post-quantum' low-dimensional generative models of their training data -- as if performing iterative Bayesian updates over the latent state of this world model during inference as they observe more context. Notably, neural nets easily find these representation whereas there is no finite classical circuit that would do the job. The corresponding geometric relationships among neural activations induced by different input sequences are found to be largely independent of neural-network architecture. Each point in this geometry corresponds to a history-induced probability density over all possible futures, and the relative displacement of these points reflects the difference in mechanism and magnitude for how these distinct pasts affect the future.
Based on 't Hooft's principle of superposition of states beyond the usual one as described by Dirac in the conventional quantum mechanics, we present a topos-theoretic formalism of quantum artificial intelligence. At first, according to Turing test, we interpret an artificial intelligence (AI) system as a physical system described by a topos (which is a new physical theory built by Isham et al. in 2008). Secondly, by using measure theory, we construct a topos-theoretic model for a classical AI system such as deep learning, while we construct a topos-theoretic model for a quantum AI system by operator theory. Finally, we give the topos-theoretic description of a quantum neural network.
The rapid commercialization of generative artificial intelligence (AI), along with the maturation of quantum technologies has raised a question: can quantum-powered neural networks become the next major shift in large language model (LLM) technology? This naturally leads to another misconception that quantum systems will replace classical LLMs. In this study, both architectures are compared in a contrastive manner in terms of mathematics. The data reveals that identical dynamics that help classical systems learn natural language distributions constrain its ability to use efficient sampling of quantum-mechanical spaces. Performing complexity-theoretic separations (i.e., the widely believed but unproven conjecture that BPP ⊆ BQP) and a 2025 preprint reporting experimental demonstrations of quantum advantage for generative tasks we conclude that quantum utility is unlikely to lie in tasks involving natural language processing under current architectures, but rather in certain computational subroutines. We then suggest a hybrid quantum-classical architecture as the best direction to take in the future, as it has the advantages of both paradigms. This is done by studying a case study that optimizes retrieval-augmented generation (RAG) pipelines with Grover's search algorithm.
Quantum Computing and Generative Artificial Intelligence are transforming the future of intelligent computing systems.The integration of quantum computing with generative AI introduces a new paradigm called Quantum-Enhanced Generative AI (QEGAI), which utilizes quantum mechanical principles such as superposition, entanglement, and quantum parallelism to improve computational efficiency, model training, optimization, and intelligent decision-making.Classical generative AI models such as Generative Adversarial Networks (GANs), Variational Autoencoders (VAEs), and Large Language Models (LLMs) demand extensive computational resources and energy consumption.Quantum-enhanced techniques address these limitations by accelerating optimization and enabling faster processing of complex datasets.This paper presents an in-depth study of QEGAI architectures, working principles, applications, advantages, limitations, and future research directions.The proposed framework combines quantum processors with classical AI infrastructure to create scalable and intelligent next-generation computing environments.The study also explores the role of quantum machine learning, hybrid quantum-classical architectures, cybersecurity, healthcare, smart cities, and scientific simulations.Comparative analysis demonstrates the superior computational capabilities of QEGAI over traditional AI approaches in selected tasks.Finally, the paper discusses implementation challenges such as qubit instability, noise, hardware scalability, and ethical concerns while outlining future opportunities for sustainable intelligent computing systems.
Generative AI has emerged as a transformative paradigm for diverse applications, yet the escalating scale of modern models exposes critical computational and memory bottlenecks in classical hardware. This paper investigates the intersection of quantum computing and generative artificial intelligence (QGAI) to address these limitations and scale modern generative models. As models grow to billions of parameters, classical systems face bottlenecks in memory, energy, and training efficiency, while quantum computing offers exponential representational benefits for high-dimensional data. The paper analyzes five core quantum generative architectures-Quantum Circuit Born Machines, Quantum Generative Adversarial Networks, Quantum Boltzmann Machines, Quantum Variational Autoencoders, and Quantum Diffusion models, highlighting their design principles, learning mechanisms, and applications. QGAI models have demonstrated significant promise in domains such as drug discovery, human-machine interaction, IoT security, and financial modelling. Despite these advances, QGAI remains constrained by qubit noise, barren plateaus, and integration challenges. We conclude by identifying ten open research challenges and propose directions for achieving scalable, interpretable, and energy-efficient quantum generative learning.