Quantum Maximum Likelihood Prediction via Hilbert Space Embeddings

2026-02-20

AI Summary

This work proposes a unified theoretical framework to elucidate the prediction mechanism of large language models in in-context learning. By adopting an information-geometric and statistical perspective, it interprets model training as an embedding of probability distributions into the space of quantum density operators and formulates in-context learning as maximum likelihood prediction over a specific class of quantum models. Leveraging quantum reverse information projection and the quantum Pythagorean theorem—combined with Hilbert space embeddings, quantum relative entropy, and non-asymptotic analysis—the study derives convergence rates and concentration inequalities in trace norm. This framework coherently characterizes both classical and quantum large language models, offering the first non-asymptotic performance guarantees for in-context learning grounded in quantum information geometry.

Technology Category

• Machine Learning: Quantum Machine Learning
• Natural Language Processing: (Large) Language Models
• Search and Optimization: Learning to Search

Application Category

• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Recent works have proposed various explanations for the ability of modern large language models (LLMs) to perform in-context prediction. We propose an alternative conceptual viewpoint from an information-geometric and statistical perspective. Motivated by Bach[2023], we model training as learning an embedding of probability distributions into the space of quantum density operators, and in-context learning as maximum-likelihood prediction over a specified class of quantum models. We provide an interpretation of this predictor in terms of quantum reverse information projection and quantum Pythagorean theorem when the class of quantum models is sufficiently expressive. We further derive non-asymptotic performance guarantees in terms of convergence rates and concentration inequalities, both in trace norm and quantum relative entropy. Our approach provides a unified framework to handle both classical and quantum LLMs.

Problem

• in-context learning
• large language models
• quantum maximum likelihood
• information geometry
• Hilbert space embeddings

Innovation

• quantum embedding
• maximum likelihood prediction
• in-context learning
• information geometry
• quantum relative entropy