用量子嵌入方法实现更优的极大似然预测，统一经典与量子大模型框架。
Quantum Maximum Likelihood Prediction via Hilbert Space Embeddings
- 将经验分布嵌入量子态，通过最小化量子相对熵求解预测
- 给出非渐近收敛速率与浓度不等式，覆盖迹范数与相对熵两种度量
- 推广量子勾股定理至非自伴混合族，适用于无限维情形
极大似然预测（MLP）是现代大语言模型的核心任务。本文研究了一种简化数据模型下的量子版本MLP，该模型由独立同分布样本构成。量子极大似然预测器（QMLP）通过将经验概率分布嵌入量子态，并在给定状态类上最小化量子相对熵获得。我们推导了QMLP在迹范数和量子相对熵下的非渐近性能保证，包括收敛速率与浓度不等式。该方法为经典与量子大语言模型中的MLP提供了统一框架。此外，我们还研究了量子信息投影问题，将经典的量子勾股定理推广至非自伴生成的混合族，并证明在附加正则性条件下，勾股不等式在无限维情形下依然成立。
原文摘要 · Abstract (English)
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 well known quantum Pythagorean theorem to mixture families which are not necessarily generated by a self-adjoint class. We further show that the Pythagorean inequality continues to hold in the infinite dimensional setting under additional regularity conditions.
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