量子近似计数与伯努利预言机

中文摘要 AI 辅助

量子计数是一种基本的量子算法，它利用成员资格预言机来估计标记元素的比例，相较于经典采样实现了二次加速。然而，成员资格预言机假设每个元素都被精确标记，但当标记本身具有概率性时，这一假设便不再成立。我们研究了使用伯努利预言机的量子计数问题，其中给定m个具有未知偏差p_1,…,p_m的伯努利分布以及一个间隙参数Δ，目标是估计“正”分布（即p_i≥1/2+Δ）的比例ρ，误差控制在加法误差ε以内。我们证明了查询复杂度的上界为Õ(√ρ/(Δε)+1/(Δ√ε))，相较于经典样本复杂度实现了二次加速。我们的算法首先利用量子奇异值变换（QSVT）来相干地放大偏差间隙，而不破坏对分布的叠加态，然后应用两阶段自适应振幅估计。我们通过一个新的量子对手方法在布尔平均情况方向上的组合定理，补充了一个近乎匹配的下界Ω(√ρ/(Δε))。对于常数间隙Δ=Θ(1)的特殊情况，这对应于有界误差预言机，即每次查询以常数概率返回正确标签，我们的界分别简化为Õ(√ρ/ε+1/√ε)和Ω(√ρ/ε)，从而刻画了有界误差预言机下量子计数的查询复杂度。

英文摘要

Quantum counting is a fundamental quantum algorithm that estimates the fraction of marked elements using a membership oracle, achieving a quadratic speedup over classical sampling. The membership oracle, however, assumes exact labeling of each element, but this assumption fails when the labels are inherently probabilistic. We study quantum counting with \emph{Bernoulli oracles}, where given $m$ Bernoulli distributions with unknown biases $p_1,\dots,p_m$ and a gap parameter $Δ$, the goal is to estimate the fraction $ρ$ of \emph{positive} distributions ($p_i\ge1/2+Δ$) to within additive error $ε$. We prove an upper bound of $\tilde{O}\!\big(\frac{\sqrtρ}{Δε}+\frac{1}{Δ\sqrtε}\big)$ queries, achieving a quadratic speedup over the classical sample complexity. % of $Θ(ρ/Δ^2ε^2)$. Our algorithm first uses the Quantum Singular Value Transformation (QSVT) to coherently amplify the bias gap without collapsing the superposition over distributions, and then applies two-stage adaptive amplitude estimation. We complement this upper bound with a near-matching lower bound of $Ω(\sqrtρ/(Δε))$ via a new composition theorem for the quantum adversary method in the Boolean-over-average-case direction. For the special case of a constant gap $Δ=Θ(1)$, which corresponds to the bounded-error oracle where each query returns the correct label with constant probability, our bounds specialize to $\tilde{O}\big(\frac{\sqrtρ}ε+\frac{1}{\sqrtε}\big)$ and $Ω(\frac{\sqrtρ}ε)$, thereby characterizing the query complexity of quantum counting with bounded-error oracles.

发表机构

• Sun Yat-sen University（中山大学）
• Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area（粤港澳大湾区量子科学中心）

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