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Certified Lower Bounds and Efficient Estimation of Minimum Accuracy in Quantum Kernel Methods

Demerson N. Gonçalves, Tharso D. Fernandes, Andrias M. M. Cordeiro, Pedro H. G. Lugao, João T. Dias

TL;DR

This work generalizes the minimum accuracy metric to arbitrary binary datasets and proves it is a certified lower bound on the best empirical accuracy $R^*$ of any linear classifier in the quantum feature space, i.e., $R_{ m min} \le R^*$. It then introduces Monte Carlo axis-selection methods with probabilistic guarantees to estimate $R_{ m min}$ from a small subset of Pauli directions, dramatically reducing computational cost. The proposed framework lowers the QSVM design burden from exponential to near-linear in the number of evaluated axes, while providing rigorous bounds and interpretability. Experimental results on synthetic data show tight lower bounds with substantial speedups, enabling effective pre-screening of quantum feature maps for near-term quantum devices.

Abstract

The minimum accuracy heuristic evaluates quantum feature maps without requiring full quantum support vector machine (QSVM) training. However, the original formulation is computationally expensive, restricted to balanced datasets, and lacks theoretical backing. This work generalizes the metric to arbitrary binary datasets and formally proves it constitutes a certified lower bound on the optimal empirical accuracy of any linear classifier in the same feature space. Furthermore, we introduce Monte Carlo strategies to efficiently estimate this bound using a random subset of Pauli directions, accompanied by rigorous probabilistic guarantees. These contributions establish minimum accuracy as a scalable, theoretically sound tool for pre-screening feature maps on near-term quantum devices.

Certified Lower Bounds and Efficient Estimation of Minimum Accuracy in Quantum Kernel Methods

TL;DR

This work generalizes the minimum accuracy metric to arbitrary binary datasets and proves it is a certified lower bound on the best empirical accuracy of any linear classifier in the quantum feature space, i.e., . It then introduces Monte Carlo axis-selection methods with probabilistic guarantees to estimate from a small subset of Pauli directions, dramatically reducing computational cost. The proposed framework lowers the QSVM design burden from exponential to near-linear in the number of evaluated axes, while providing rigorous bounds and interpretability. Experimental results on synthetic data show tight lower bounds with substantial speedups, enabling effective pre-screening of quantum feature maps for near-term quantum devices.

Abstract

The minimum accuracy heuristic evaluates quantum feature maps without requiring full quantum support vector machine (QSVM) training. However, the original formulation is computationally expensive, restricted to balanced datasets, and lacks theoretical backing. This work generalizes the metric to arbitrary binary datasets and formally proves it constitutes a certified lower bound on the optimal empirical accuracy of any linear classifier in the same feature space. Furthermore, we introduce Monte Carlo strategies to efficiently estimate this bound using a random subset of Pauli directions, accompanied by rigorous probabilistic guarantees. These contributions establish minimum accuracy as a scalable, theoretically sound tool for pre-screening feature maps on near-term quantum devices.
Paper Structure (16 sections, 4 theorems, 20 equations, 2 figures, 1 table)

This paper contains 16 sections, 4 theorems, 20 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Generalized minimum accuracy
  3. Minimum accuracy as a lower bound on optimal linear classifiers
  4. Monte Carlo axis selection with statistical guarantees
  5. Estimator definition and basic properties
  6. Quantile coverage guarantees
  7. Monte Carlo Sampling Methods for Axis Selection
  8. Deterministic exhaustive method
  9. Conservative fixed-sample Monte Carlo
  10. Pilot sampling Monte Carlo
  11. Adaptive incremental Monte Carlo
  12. Experimental Results
  13. Experimental Setup and Feature Construction
  14. Sensitivity Analysis and Performance Results
  15. Discussion
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\Phi : \mathcal{X} \to \mathbb{R}^d$ be a feature map and let $D = \{(x_k, y_k)\}_{k=1}^N$ be a binary labeled dataset. Let $\mathcal{F}$ be the class of all linear classifiers in the feature space as defined above, with optimal empirical accuracy $R^*$ given by eq:R-star-def. Let $R_{\min}$ de

Figures (2)

  • Figure 1: Training accuracy for varying prior assumptions $p$. From top to bottom: $p=0.05$ (Conservative $t=60$), $p=0.15$ (Conservative $t=20$), and $p=0.25$ (Conservative $t=12$).
  • Figure 2: Number of axes sampled for different values of the prior parameter $p$. The Conservative method (orange) reduces its sampling budget as $p$ increases, while Pilot and Adaptive methods remain unaffected by this prior.

Theorems & Definitions (8)

  • Theorem 1: Minimum accuracy lower bound
  • proof
  • Theorem 2: Monte Carlo lower bound property
  • proof
  • Theorem 3: Quantile coverage
  • proof
  • Remark 1
  • Corollary 1: Sample size for accuracy threshold