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

• URL: http://arxiv.org/abs/2405.12838v1
• Date: Tue, 21 May 2024 14:42:39 GMT
• Title: Quantum Non-Identical Mean Estimation: Efficient Algorithms and Fundamental Limits
• Authors: Jiachen Hu, Tongyang Li, Xinzhao Wang, Yecheng Xue, Chenyi Zhang, Han Zhong,
• Abstract summary: We 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.
• Score: 15.89518426969296
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\mu$, where the samples come from different random variables with mean close to $\mu$. 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.

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