Quantum Multi-Level Estimation of Functionals of Discrete Distributions

TL;DR

This paper introduces a quantum multi-level estimation framework for functionals of discrete distributions, achieving near-optimal quantum algorithms for estimating $q$-Tsallis entropy with improved query complexities.

Contribution

The authors develop a novel quantum estimation method that partitions probability values into intervals, enabling efficient estimation of functionals like $q$-Tsallis entropy with better query complexity bounds.

Findings

Quantum algorithms for $q>1$ with near-optimal query complexity.

Quantum estimators for $0<q<1$ showing quantum speedup.

First near-optimal quantum estimators for non-integer $q$-entropy.

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 $\tilde{\Theta}(1/\varepsilon^{\max\{1/(2(q-1)), 1\}})$, 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 $q$.