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Quantum Multi-Level Estimation of Functionals of Discrete Distributions

Authors Kean Chen , Minbo Gao , Tongyang Li , Qisheng Wang , Xinzhao Wang

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LIPIcs.ICALP.2026.58.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Information theory
Keywords
  • Quantum algorithms
  • functional estimation
  • entropy estimation
  • query complexity
  • Tsallis entropy

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Abstract

We propose a quantum multi-level estimation framework for a functional ∑_{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,  
- For q > 1, we obtain a near-optimal quantum algorithm with query complexity Θ̃(1/ε^{max{1/(2(q-1)), 1}}), improving the prior best O(1/ε^{1+1/(q-1)}) due to Liu and Wang (SODA 2025; IEEE Trans. Inf. Theory 2026). 
- For 0 < q < 1, we obtain a quantum algorithm with query complexity Õ(n^{1/q-1/2}/ε^{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.

Cite As Get BibTex

Kean Chen, Minbo Gao, Tongyang Li, Qisheng Wang, and Xinzhao Wang. Quantum Multi-Level Estimation of Functionals of Discrete Distributions. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 58:1-58:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.58

Author Details

Kean Chen
  • Department of Computer and Information Science, University of Pennsylvania, Philadelphia, PA, USA
Minbo Gao
  • Key Laboratory of System Software (Chinese Academy of Sciences), Institute of Software, Chinese Academy of Sciences, Beijing, China
  • University of Chinese Academy of Sciences, Beijing, China
Tongyang Li
  • Center on Frontiers of Computing Studies, Peking University, Beijing, China
  • School of Computer Science, Peking University, Beijing, China
Qisheng Wang
  • School of Computer Science, Shanghai Jiao Tong University, Shanghai, China
Xinzhao Wang
  • Center on Frontiers of Computing Studies, Peking University, Beijing, China
  • School of Computer Science, Peking University, Beijing, China

Funding

  • Gao, Minbo: Supported by National Key Research and Development Program of China (Grant No. 2023YFA1009403).
  • Li, Tongyang: Supported by the National Natural Science Foundation of China (Grant No. 62372006).
  • Wang, Qisheng: Supported by startup funding from Shanghai Jiao Tong University.
  • Wang, Xinzhao: Supported by the National Natural Science Foundation of China (Grant No. 62372006).

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