Nearly Sample-Optimal Estimators for Quantum Rényi and Tsallis Entropies

In this paper, we provide estimators for quantum Rényi and Tsallis entropies with nearly optimal sample complexity. Specifically, for order $\alpha$, dimension $d$, and additive error $\epsilon$, 1. For $0<\alpha <1$, the sample complexity is $O(d^{1+1\mathrm{/}\alpha }\mathrm{/}{\epsilon }^{1\mathrm{/}\alpha }+d^{1\mathrm{/}\alpha -1}\mathrm{/}{\epsilon }^2)$ for Rényi entropy and $O(d^{1+1\mathrm{/}\alpha }\mathrm{/}{\epsilon }^{1\mathrm{/}\alpha }+d^{2-2\alpha }\mathrm{/}{\epsilon }^2)$ for Tsallis entropy. In particular, for $0<\alpha \le 1\mathrm{/}2$, the sample complexity for both entropies is $O(d^{1+1\mathrm{/}\alpha }\mathrm{/}{\epsilon }^{1\mathrm{/}\alpha })$. 2. For non-integer $\alpha >1$, the sample complexity is $O(d^2\mathrm{/}{\epsilon }^{1\mathrm{/}\alpha }+d^{1-1\mathrm{/}\alpha }\mathrm{/}{\epsilon }^2)$ for Rényi entropy. Our upper bounds improve the quantum Rényi entropy estimators due to Acharya, Issa, Shende, and Wagner (2017) and the quantum Tsallis entropy estimators due to Chen, Liu, and Wang (2026), and match the lower bounds recently established by Wang (2026).