# Quantum Maximum Likelihood Prediction via Hilbert Space Embeddings

arXiv: https://arxiv.org/abs/2602.18364 · HTML: https://mlsanity.com/paper/2602.18364/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 80 references that are in the map.
- First posted on arXiv 20 Feb 2026.

## Builds on (most cited first, 22 of 22)

- [An Explanation of In-context Learning as Implicit Bayesian Inference](https://mlsanity.com/paper/2111.02080/) (2021)
- [What Can Transformers Learn In-Context? A Case Study of Simple Function Classes](https://mlsanity.com/paper/2208.01066/) (2022)
- [What learning algorithm is in-context learning? Investigations with linear models](https://mlsanity.com/paper/2211.15661/) (2022)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Transformers as Statisticians: Provable In-Context Learning with In-Context Algorithm Selection](https://mlsanity.com/paper/2306.04637/) (2023)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [The Evolution of Statistical Induction Heads: In-Context Learning Markov Chains](https://mlsanity.com/paper/2402.11004/) (2024)
- [An Information-Theoretic Analysis of In-Context Learning](https://mlsanity.com/paper/2401.15530/) (2024)
- [Information Theory with Kernel Methods](https://mlsanity.com/paper/2202.08545/) (2022)
- [Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Models](https://mlsanity.com/paper/1609.02542/) (2016)
- [Understanding LLM Behaviors via Compression: Data Generation, Knowledge Acquisition and Scaling Laws](https://mlsanity.com/paper/2504.09597/) (2025)
- [Towards Understanding the Universality of Transformers for Next-Token Prediction](https://mlsanity.com/paper/2410.03011/) (2024)
- [Semidefinite approximations of the matrix logarithm](https://mlsanity.com/paper/1705.00812/) (2017)
- [Optimal Estimation of Low Rank Density Matrices](https://mlsanity.com/paper/1507.05131/) (2015)
- [Minimizing Quantum Renyi Divergences via Mirror Descent with Polyak Step Size](https://mlsanity.com/paper/2109.06054/) (2021)
- [Learning to Embed Distributions via Maximum Kernel Entropy](https://mlsanity.com/paper/2408.00549/) (2024)
- [Performance Guarantees for Quantum Neural Estimation of Entropies](https://mlsanity.com/paper/2511.19289/) (2025)
- [Fundamentals of quantum Boltzmann machine learning with visible and hidden units](https://mlsanity.com/paper/2512.19819/) (2025)
- [Optimal self-concordant barriers for quantum relative entropies](https://mlsanity.com/paper/2205.04581/) (2022)
- [Transformers are Universal Predictors](https://mlsanity.com/paper/2307.07843/) (2023)
- [Interior Point Methods for Structured Quantum Relative Entropy Optimization Problems](https://mlsanity.com/paper/2407.00241/) (2024)
- [Likelihood Ratio Tests by Kernel Gaussian Embedding](https://mlsanity.com/paper/2508.07982/) (2025)

## Abstract (arXiv)

Maximum likelihood prediction (MLP) is a core task at the heart of modern large language models. Here, we study a quantum version of this task for a simplified data model consisting of independent and identically distributed samples, as a first step. The quantum maximum likelihood predictor (QMLP) is obtained by embedding of empirical probability distributions into quantum states and performing a minimization of quantum relative entropy over a given class of states. We derive non-asymptotic performance guarantees for QMLP in terms of convergence rates and concentration inequalities, both in trace norm and quantum relative entropy. Our approach provides a unified framework to handle MLP within both classical and quantum LLMs. We also consider the related problem of quantum information projection and generalize the quantum Pythagorean theorem to mixture families specified by possibly non-self-adjoint linear constraints. We further show that the Pythagorean inequality continues to hold in the infinite-dimensional setting whenever the convex information-projection problem attains a finite minimum.