基于最优量子Tsallis熵估计的单项式近似的信息论下界
Information-Theoretic Lower Bounds for Approximating Monomials via Optimal Quantum Tsallis Entropy Estimation
AI总结：
该研究建立单项式多项式近似与量子Tsallis熵估计的关联，得到单项式近似度的信息论下界，还提出查询复杂度更优的量子Tsallis熵估计算法，是首个全参数同时达最优（至多差多对数因子）的量子熵估计器。
摘要：
本文通过熵估计算法揭示了信息论与近似理论之间一个概念上的全新关联。具体而言，我们给出了单项式$x^n$近似度的信息论下界$Ω(\sqrt{n})$，这一结果可与Newman和Rivlin（《Aequ. Math.》1976年）通过傅里叶分析、以及Sachdeva和Vishnoi（《Found. Trends Theor. Comput. Sci.》2014年）通过马尔可夫兄弟不等式得到的解析下界相比对。该下界通过将单项式的多项式近似与量子Tsallis熵估计建立关联得到。这一关联进一步推导出一种量子算法，可对未知概率分布$p$或未知量子态$ρ$的整数阶$q≥2$的Tsallis熵进行加性误差$ε$范围内的估计，仅需$\widetilde Θ(\frac{1}{\sqrt{q}\varepsilon})$次对生成$p$样本或制备$ρ$副本的量子预言机的查询，优于此前由Ekert、Alves、Oi、Horodecki、Horodecki和Kwek（《Phys. Rev. Lett.》2002年）提出的Shift测试方法的$O(\frac{1}{\varepsilon})$查询复杂度。据我们所知，这是首个在所有参数上同时达到最优查询复杂度（至多相差多对数因子）的量子熵估计器。
英文摘要：
This paper reveals a conceptually new connection from information theory to approximation theory via quantum algorithms for entropy estimation. Specifically, we provide an information-theoretic lower bound $Ω(\sqrt{n})$ on the approximate degree of the monomial $x^n$, compared to the analytic lower bounds shown in Newman and Rivlin (Aequ. Math. 1976) via Fourier analysis and in Sachdeva and Vishnoi (Found. Trends Theor. Comput. Sci. 2014) via the Markov brothers' inequality. This is done by relating the polynomial approximation of monomials to quantum Tsallis entropy estimation. This further implies a quantum algorithm that estimates to within additive error $\varepsilon$ the Tsallis entropy of integer order $q \geq 2$ of an unknown probability distribution $p$ or an unknown quantum state $ρ$, using $\widetilde Θ(\frac{1}{\sqrt{q}\varepsilon})$ queries to the quantum oracle that produces a sample from $p$ or prepares a copy of $ρ$, improving the prior best $O(\frac{1}{\varepsilon})$ via the Shift test due to Ekert, Alves, Oi, Horodecki, Horodecki and Kwek (Phys. Rev. Lett. 2002). To the best of our knowledge, this is the first quantum entropy estimator with optimal query complexity (up to polylogarithmic factors) for all parameters simultaneously.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
