Quantum Statistical Bootstrap

2026-04-01

AI Summary

This work addresses the high approximation error and substantial computational cost inherent in classical bootstrapping, which relies on Monte Carlo resampling—issues that are exacerbated with large datasets and complex models. The authors propose Quantum Bootstrap (QBOOT), a novel approach that leverages quantum superposition to encode all resamples simultaneously, enabling parallel evaluation of the target statistic within a quantum circuit. By employing quantum amplitude estimation, QBOOT precisely extracts aggregated results, achieving the first unbiased quantum computation of the ideal bootstrap estimator. Under mild assumptions on circuit efficiency, it attains a near-quadratic speedup independent of the specific statistic or underlying distribution, while preserving the asymptotic properties of classical bootstrapping. Experiments on IBM quantum simulators demonstrate its high accuracy and significant acceleration in inferring sample means, thereby filling a critical theoretical gap in statistical error analysis for quantum algorithms.

Abstract

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.

Problem

• bootstrap
• statistical inference
• computational cost
• approximation error
• quantum algorithms

Innovation

• Quantum Bootstrap
• quantum amplitude estimation
• statistical inference
• quantum speedup
• resampling