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Quantum Non-Identical Mean Estimation: Efficient Algorithms and Fundamental Limits

Authors Jiachen Hu , Tongyang Li , Xinzhao Wang , Yecheng Xue , Chenyi Zhang , Han Zhong

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Author Details

Jiachen Hu
  • Peking University, Beijing, China
Tongyang Li
  • Peking University, Beijing, China
Xinzhao Wang
  • Peking University, Beijing, China
Yecheng Xue
  • Peking University, Beijing, China
Chenyi Zhang
  • Stanford University, CA, USA
Han Zhong
  • Peking University, Beijing, China

Funding

TL, XW, and YX were supported by the National Natural Science Foundation of China (Grant Numbers 62372006 and 92365117), and The Fundamental Research Funds for the Central Universities, Peking University.

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Jiachen Hu, Tongyang Li, Xinzhao Wang, Yecheng Xue, Chenyi Zhang, and Han Zhong. Quantum Non-Identical Mean Estimation: Efficient Algorithms and Fundamental Limits. In 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 310, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.TQC.2024.9

Abstract

We systematically investigate quantum algorithms and lower bounds for mean estimation given query access to non-identically distributed samples. On the one hand, we give quantum mean estimators with quadratic quantum speed-up given samples from different bounded or sub-Gaussian random variables. On the other hand, we prove that, in general, it is impossible for any quantum algorithm to achieve quadratic speed-up over the number of classical samples needed to estimate the mean μ, where the samples come from different random variables with mean close to μ. Technically, our quantum algorithms reduce bounded and sub-Gaussian random variables to the Bernoulli case, and use an uncomputation trick to overcome the challenge that direct amplitude estimation does not work with non-identical query access. Our quantum query lower bounds are established by simulating non-identical oracles by parallel oracles, and also by an adversarial method with non-identical oracles. Both results pave the way for proving quantum query lower bounds with non-identical oracles in general, which may be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
Keywords
  • Quantum algorithms
  • Mean estimation
  • Non-identical samples
  • Query complexity

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