The bootstrap is a foundational tool in statistical inference, but its classical implementation relies on Monte Carlo resampling, introducing approximation error and incurring high computational cost -- especially for large datasets and complex models. We present the Quantum Bootstrap (QBOOT), a quantum algorithm that computes the ideal bootstrap estimate exactly by encoding all possible resamples in quantum superposition, evaluating the target statistic in parallel, and extracting the aggregate via quantum amplitude estimation. Under mild circuit efficiency assumptions, QBOOT achieves a near-quadratic speedup over the classical bootstrap in approximating the ideal estimator, independent of the statistic or underlying distribution. We provide a rigorous theoretical analysis of its statistical error properties -- addressing a gap in the quantum algorithms literature -- and validate our results through experiments on the IBM quantum simulator for the sample mean problem. Our findings demonstrate that QBOOT preserves the asymptotic properties of the ideal bootstrap while substantially improving computational efficiency and precision, establishing a scalable and principled framework for quantum statistical inference.

翻译：自助法（Bootstrap）是统计推断中的基础工具，但其经典实现依赖于蒙特卡洛重采样，会引入近似误差并产生高昂的计算成本——尤其是在处理大规模数据集和复杂模型时。我们提出量子自助法（QBOOT），这是一种量子算法，通过将所有可能的子样本编码为量子叠加态、并行评估目标统计量，并借助量子振幅估计提取聚合结果，从而精确计算理想自助估计量。在适度的电路效率假设下，QBOOT在逼近理想估计量时相较经典自助法实现了近二次加速，且此加速与统计量或底层分布无关。我们对其统计误差特性进行了严格的理论分析——填补了量子算法文献中的一项空白——并通过IBM量子模拟器在样本均值问题上的实验验证了我们的结果。研究表明，QBOOT在显著提升计算效率与精度的同时，保留了理想自助法的渐近性质，为量子统计推断建立了可扩展且原理性强的框架。