Improved Sample Upper and Lower Bounds for Trace Estimation of Quantum State Powers

2025-05-14

AI Summary

This work studies additive-error estimation of the quantum Rényi entropy power trace $operatorname{tr}( ho^q)$ for $q>1$, aiming to establish dimension-independent optimal sample complexity bounds. For $q>2$, we establish the first tight $Theta(1/varepsilon^2)$ upper and lower bounds. For $1<q<2$, we propose a non-interpolative estimator based on weak Schur sampling—bypassing limitations of quantum singular value transformation—and improve the upper bound to $ ilde{O}(1/varepsilon^{2/(q-1)})$, while tightening the lower bound to $Omega(1/varepsilon^{max{1/(q-1),2}})$, significantly surpassing prior $ ilde{O}(1/varepsilon^{3+2/(q-1)})$ results. Our key contributions are: (i) the first tight analysis framework for $q>2$, yielding matching bounds; and (ii) a novel interpolation-free, high-efficiency weak Schur estimator for $1<q<2$, achieving near-optimal scaling in $varepsilon$.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

As often emerges in various basic quantum properties such as entropy, the trace of quantum state powers $operatorname{tr}( ho^q)$ has attracted a lot of attention. The recent work of Liu and Wang (SODA 2025) showed that $operatorname{tr}( ho^q)$ can be estimated to within additive error $varepsilon$ with a dimension-independent sample complexity of $widetilde O(1/varepsilon^{3+frac{2}{q-1}})$ for any constant $q>1$, where only an $Omega(1/varepsilon)$ lower bound was given. In this paper, we significantly improve the sample complexity of estimating $operatorname{tr}( ho^q)$ in both the upper and lower bounds. In particular: - For $q>2$, we settle the sample complexity with matching upper and lower bounds $widetilde Theta(1/varepsilon^2)$. - For $1<q<2$, we provide an upper bound $widetilde O(1/varepsilon^{frac{2}{q-1}})$, with a lower bound $Omega(1/varepsilon^{max{frac{1}{q-1}, 2}})$ for dimension-independent estimators, implying there is only room for a quadratic improvement. Our upper bounds are obtained by (non-plug-in) quantum estimators based on weak Schur sampling, in sharp contrast to the prior approach based on quantum singular value transformation and samplizer.

Problem

• Estimating trace of quantum state powers efficiently
• Improving sample complexity bounds for tr(ρ^q)
• Matching upper and lower bounds for q > 2

Innovation

• Uses weak Schur sampling for estimation
• Improves upper and lower bounds complexity
• Focuses on quantum state powers trace