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Quantum Statistical Bootstrap

Yongkai Chen, Ping Ma, Wenxuan Zhong

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.

Quantum Statistical Bootstrap

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.

Paper Structure

This paper contains 51 sections, 8 theorems, 68 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. QBOOT in brief.
  3. Contributions.
  4. Results
  5. Preliminary
  6. Ideal nonparametric bootstrap.
  7. Monte Carlo (classical) bootstrap approximation.
  8. Computational complexity of the classical bootstrap.
  9. Quantum Bootstrap (QBOOT)
  10. State encoding.
  11. Coherent evaluation of the indicator.
  12. Amplitude estimation and read‑out.
  13. Multiple bootstrap CDF values via parallelism.
  14. Theoretical Results
  15. Quantum error.
  16. ...and 36 more sections

Key Result

Theorem 2.1

Fix $T\in\mathbb{N}$ and $z\in\mathbb{R}$. Conditional on $H_{\mathrm{BOOT}}(z)$ (equivalently, on the observed data), the QBOOT estimator admits the exact representation where $Y_z\in\{0,1,\dots,2^T-1\}$ has probability mass function with $\tau_z = \arcsin\sqrt{H_{\mathrm{BOOT}}(z)}$ for $\tau_z \notin \{\frac{m\pi}{2^T}: m\in\mathbb{Z}\}$, with eq:exact_pmf understood by continuous extension a

Figures (5)

  • Figure 1: Bias, variance, and MSE of $H_{\mathrm{QBOOT}}(z)$ as functions of $H_{\mathrm{BOOT}}(z)$ for $T\in\{2,4,6,8\}$, showing oscillations and the vanishing of error when $2^T\tau_z/\pi\in\mathbb{Z}$.
  • Figure 2: QBOOT vs. CBOOT at matched cost. (a) Boxplots of $\log$ absolute errors vs. $T$, with theoretical guides $O(2^{-T})$ (QBOOT, red) and $O(2^{-T/2})$ (CBOOT, blue). (b) MAE, $\sqrt{\mathrm{MSE}}$, and median absolute error vs. $T$.
  • Figure 3: Robust aggregation for QBOOT. (a) Absolute error boxplots vs. $T$ for $M\in\{1,3,5,7\}$ repeated QAE measurements; the median across $M$ runs is reported per replication. (b) $\sqrt{\mathrm{MSE}}$ vs. $T$; robust aggregation (e.g., $M=5$) closely tracks the $O(2^{-T})$ guide.
  • Figure S.1: Probability distribution of QAE outcomes $\mathop{\rm Pr}\nolimits(l_z=l)$ for $\tau_z=0.5$ and $T\in\{4,8,12\}$, illustrating concentration near $2^{-T}l\pi\approx \tau_z$ that drives the $\mathcal{O}_p(2^{-T})$ error bound.
  • Figure S.2: Example realisation of $\mathbf{U}_g(z)$ for the sample‑mean indicator $g(z,\boldsymbol{X}^*)=\mathbf{1}\{\bar{X}^*\le z\}$. The circuit accumulates $\sum_j X_j^*$ in a statistic register, compares to $z$, flips a label qubit, and uncomputes the arithmetic.

Theorems & Definitions (9)

  • Theorem 2.1: Exact distribution of QBOOT
  • Theorem 2.2: Quantum error stochastic bound
  • Theorem 2.3: Bias and MSE
  • Corollary 2.4: Uniform asymptotics
  • Theorem 2.5: QBOOT gate complexity
  • Lemma 2.6: Single-run error bound
  • Theorem 2.7: Median-of-$M$ error bound
  • Theorem 2.8: Median estimate preserves the quadratic speedup
  • Remark 2.9