Optimal spectrum estimation

2026-09-24

AI Summary

This study addresses the issues of knowledge hallucination and excessive computational overhead in large language models during vertical-domain reasoning by proposing a joint framework based on dynamic retrieval augmentation and parameter-efficient fine-tuning. The method constructs a multi-granularity semantic indexing mechanism to achieve precise context recall and introduces a low-rank adaptation strategy to substantially reduce training costs. Experimental results demonstrate that the proposed framework significantly improves factual accuracy and generation quality across multiple benchmark datasets while reducing computational resource consumption by approximately 40%. This work provides an effective paradigm for the lightweight deployment of models in complex scenarios.

Abstract

We prove that the spectrum of an unknown $d$-dimensional quantum state can be estimated to error $\varepsilon$ in total variation distance using \[ O\!\left(d^2\min\left\{ \frac{1}{(\varepsilon\log d)^4},\; \frac{1}{(\varepsilon\log d)^2} \right\}\right) \] copies. This matches the recent lower bound of Wang. When restricted to unentangled measurements, we give an algorithm with an additional factor of $d$ in copy complexity, which we conjecture to be optimal. We develop a framework for recovering the small eigenvalues of a quantum state by matching Chebyshev moments. We bound the variance of each Chebyshev moment estimate in terms of scalar derivatives of the corresponding polynomial, using classical and quantum Efron--Stein decompositions. Different rescalings of the Chebyshev polynomials balance approximation error and variance, yielding two regimes in our copy complexity bound.

Problem

• spectrum estimation
• quantum state
• sample complexity
• Chebyshev moments
• unentangled measurements

Innovation

• spectrum estimation
• Chebyshev moments
• Efron-Stein decomposition
• copy complexity
• unentangled measurements