Quantum Multi-Level Estimation of Functionals of Discrete Distributions

2026-05-05

AI Summary

Existing quantum estimation algorithms struggle to efficiently compute non-integer-order discrete distribution functionals—such as the q-Tsallis entropy—with near-optimal query complexity. This work proposes a quantum multilevel estimation framework that achieves efficient reconstruction of functional partial sums by partitioning probability values into exponentially decaying logarithmic intervals and integrating lossless singular value discrimination with adaptive amplitude estimation. The method yields, for the first time, near-optimal quantum estimation of q-Tsallis entropy for non-integer q: for q > 1, it attains query complexity ~Θ(1/ε^{max{1/(2(q−1)),1}}), improving upon the prior O(1/ε^{1+1/(q−1)}); for 0 < q < 1, it achieves ~O(n^{1/q−1/2}/ε^{1/q}), demonstrating quantum speedup beyond classical lower bounds. Notably, the algorithm requires only a constant number of ancillary qubits, avoiding high control overhead.

Abstract

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 $q$.

Problem

• quantum estimation
• discrete distributions
• Tsallis entropy
• functionals
• query complexity

Innovation

• quantum multi-level estimation
• non-destructive singular value discrimination
• q-Tsallis entropy
• quantum query complexity
• adaptive quantum estimation