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arXiv:2405.12838 · Leibniz international proceedings in informatics

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

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

Published Jan 1, 2024• 0 citations
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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.

DOI
10.4230/lipics.tqc.2024.9

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In this paper we provide new quantum algorithms with polynomial speed-up for a range of problems for which no such results were known, or we improve previous algorithms. First, we consider the approximation of the frequency moments FkF_kFk​ of order k≥3k \geq 3k≥3 in the multi-pass streaming model with updates (turnstile model). We design a PPP-pass quantum streaming algorithm with memory MMM satisfying a tradeoff of P2M=O~(n1−2/k)P^2 M = \tilde{O}(n^{1-2/k})P2M=O~(n1−2/k), whereas the best classical algorithm requires PM=Θ(n1−2/k)P M = Θ(n^{1-2/k})PM=Θ(n1−2/k). Then, we study the problem of estimating the number mmm of edges and the number ttt of triangles given query access to an nnn-vertex graph. We describe optimal quantum algorithms that perform O~(n/m1/4)\tilde{O}(\sqrt{n}/m^{1/4})O~(n​/m1/4) and O~(n/t1/6+m3/4/t)\tilde{O}(\sqrt{n}/t^{1/6} + m^{3/4}/\sqrt{t})O~(n​/t1/6+m3/4/t​) queries respectively. This is a quadratic speed-up compared to the classical complexity of these problems. For this purpose we develop a new quantum paradigm that we call Quantum Chebyshev's inequality. Namely we demonstrate that, in a certain model of quantum sampling, one can approximate with relative error the mean of any random variable with a number of quantum samples that is linear in the ratio of the square root of the variance to the mean. Classically the dependency is quadratic. Our algorithm subsumes a previous result of Montanaro [Mon15]. This new paradigm is based on a refinement of the Amplitude Estimation algorithm of Brassard et al. [BHMT02] and of previous quantum algorithms for the mean estimation problem. We show that this speed-up is optimal, and we identify another common model of quantum sampling where it cannot be obtained. For our applications, we also adapt the variable-time amplitude amplification technique of Ambainis [Amb10] into a variable-time amplitude estimation algorithm.
We consider the problem of estimating the expected outcomes of Monte Carlo processes whose outputs are described by multidimensional random variables. We tightly characterize the quantum query complexity of this problem for various choices of oracle access to the Monte Carlo process and various normalization conditions on its output variables and the estimates to be made. More specifically, we design quantum algorithms that provide a quadratic speed-up in query complexity with respect to the precision of the resulting estimates and show that these complexities are asymptotically optimal in each of our scenarios. Interestingly, we find that these speed-ups with respect to the precision come at the cost of a (up to exponential) slowdown with respect to the dimension of the Monte Carlo random variables, for most of the parameter settings we consider. We also describe several applications of our algorithms, notably in the field of machine learning, and discuss the implications of our results.
Aaronson and Ambainis (Theory of Computing, 2014) conjectured that quantum query algorithms admit efficient almost-everywhere classical simulation: for any TTT-query quantum algorithm, its acceptance probability can be approximated on a (1−δ)(1-δ)(1−δ) fraction of inputs, up to εεε additive error, using poly(T,1/ε,1/δ)\mathrm{poly}(T, 1/ε, 1/δ)poly(T,1/ε,1/δ) classical queries. At a high level, the conjecture suggests that exponential quantum speedups are possible only on sufficiently structured inputs. In this work, we make progress on this conjecture by proving it for quantum algorithms that make massively parallel quantum queries. In contrast, Yamakawa and Zhandry (Journal of the ACM, 2024) showed that quantum algorithms restricted to parallel queries can still achieve exponential speedups over classical algorithms for sampling problems. We establish our simulation theorem by proving the stronger statement that parallel-query quantum algorithms cannot distinguish the uniform distribution over oracles from oracles drawn from so-called "dense distributions". Our main technical contribution is a coupling theorem that relates the uniform distribution over oracles to oracles drawn from dense distributions. We further extend this approach beyond the purely parallel setting. We prove simulation theorems for quantum algorithms with constant rounds of adaptivity, and for quantum algorithms with a bounded quantum-query prefix followed by a massively parallel quantum-query stage.