We propose a quantum multi-level estimation framework for a functional $\sum_{i=1}^n f(p_i)$ of a discrete distribution $(p_i)_{i=1}^n$. We partition the values $p_i$ into logarithmically many intervals whose length decays exponentially. For each interval, we perform non-destructive singular value discrimination to isolate the relevant $p_i$, enabling adaptive estimation of the partial sum over this interval. Unlike previous variable-time approaches, our method avoids high control overhead and requires only constant extra ancilla qubits. As an application, we present efficient quantum estimators for the $q$-Tsallis entropy of discrete distributions. Specifically: (i) For $q > 1$, we obtain a near-optimal quantum algorithm with query complexity $\tildeΘ(1/\varepsilon^{\max\{1/(2(q-1)), 1\}})$, improving the prior best $O(1/\varepsilon^{1+1/(q-1)})$ due to Liu and Wang (SODA 2025; IEEE Trans. Inf. Theory 2026). (ii) For $0 < q < 1$, we obtain a quantum algorithm with query complexity $\tilde{O}(n^{1/q-1/2}/\varepsilon^{1/q})$, exhibiting a quantum speedup over the near-optimal classical estimators due to Jiao, Venkat, Han, and Weissman (IEEE Trans. Inf. Theory 2017). Our results achieve, to our knowledge, the first near-optimal quantum estimators for parameterized $q$-entropy for non-integer $q$.

翻译：我们提出一种用于离散分布 $(p_i)_{i=1}^n$ 的泛函 $\sum_{i=1}^n f(p_i)$ 的量子多层次估计框架。我们将 $p_i$ 的值划分为对数多个区间，其长度呈指数级衰减。对于每个区间，我们执行非破坏性奇异值判别以分离出相关的 $p_i$，从而能够自适应地估计该区间上的部分和。与先前的可变时间方法不同，我们的方法避免了高控制开销，并且仅需恒定的额外辅助量子比特。作为应用，我们针对离散分布的 $q$-Tsallis 熵提出了高效的量子估计器。具体而言：(i) 对于 $q > 1$，我们得到了一个近似最优的量子算法，其查询复杂度为 $\tildeΘ(1/\varepsilon^{\max\{1/(2(q-1)), 1\}})$，改进了 Liu 和 Wang 此前的最佳结果 $O(1/\varepsilon^{1+1/(q-1)})$ (SODA 2025; IEEE Trans. Inf. Theory 2026)。(ii) 对于 $0 < q < 1$，我们得到了一个查询复杂度为 $\tilde{O}(n^{1/q-1/2}/\varepsilon^{1/q})$ 的量子算法，相较于 Jiao、Venkat、Han 和 Weissman (IEEE Trans. Inf. Theory 2017) 的近似最优经典估计器展现了量子加速。据我们所知，我们的结果首次实现了对非整数 $q$ 的参数化 $q$-熵的近似最优量子估计器。