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 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 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 MLP within both classical and quantum LLMs.

翻译：最大似然预测是现代大语言模型的核心任务。本文作为初步探索，针对由独立同分布样本构成的简化数据模型，研究该任务的量子版本。量子最大似然预测器通过将经验概率分布嵌入量子态，并在给定量子态类别上最小化量子相对熵而获得。当量子模型类别具有充分表达能力时，我们给出了该预测器在量子反向信息投影与量子勾股定理框架下的物理解释。进一步，我们导出非渐近性能保证，涵盖迹范数与量子相对熵下的收敛速率及集中不等式。本方法为经典与量子大语言模型中处理最大似然预测提供了统一框架。