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Training-Free Certified Bounds for Quantum Regression: A Scalable Framework

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

TL;DR

This work introduces a training-free, certified upper bound on the optimal training error for quantum regression by leveraging axis-aligned predictors in the Pauli feature space. The key idea is that the best axis predictor provides a guaranteed upper bound on the minimum MSE achievable by any linear or kernel-based regressor on the same feature map; to scale to the exponential Pauli basis, the authors develop a Monte Carlo estimator with non-asymptotic Hoeffding guarantees and an adaptive sample-size procedure. The framework enables rapid pre-screening and comparison of quantum feature maps, diagnosing expressivity and guiding architecture selection before committing to costly training. Through synthetic and real-world datasets, the method demonstrates robustness across dense and sparse regimes and shows how the bound correlates with the performance of trained models, offering a practical diagnostic tool for QML in the NISQ era.

Abstract

We present a training-free, certified error bound for quantum regression derived directly from Pauli expectation values. Generalizing the heuristic of minimum accuracy from classification to regression, we evaluate axis-aligned predictors within the Pauli feature space. We formally prove that the optimal axis-aligned predictor constitutes a rigorous upper bound on the minimum training Mean Squared Error (MSE) attainable by any linear or kernel-based regressor defined on the same quantum feature map. Since computing this exact bound requires an intractable scan of the full Pauli basis, we introduce a Monte Carlo framework to efficiently estimate it using a tractable subset of measurement axes. We further provide non-asymptotic statistical guarantees to certify performance within a practical measurement budget. This method enables rapid comparison of quantum feature maps and early diagnosis of expressivity, allowing for the informed selection of architectures before deploying higher-complexity models.

Training-Free Certified Bounds for Quantum Regression: A Scalable Framework

TL;DR

This work introduces a training-free, certified upper bound on the optimal training error for quantum regression by leveraging axis-aligned predictors in the Pauli feature space. The key idea is that the best axis predictor provides a guaranteed upper bound on the minimum MSE achievable by any linear or kernel-based regressor on the same feature map; to scale to the exponential Pauli basis, the authors develop a Monte Carlo estimator with non-asymptotic Hoeffding guarantees and an adaptive sample-size procedure. The framework enables rapid pre-screening and comparison of quantum feature maps, diagnosing expressivity and guiding architecture selection before committing to costly training. Through synthetic and real-world datasets, the method demonstrates robustness across dense and sparse regimes and shows how the bound correlates with the performance of trained models, offering a practical diagnostic tool for QML in the NISQ era.

Abstract

We present a training-free, certified error bound for quantum regression derived directly from Pauli expectation values. Generalizing the heuristic of minimum accuracy from classification to regression, we evaluate axis-aligned predictors within the Pauli feature space. We formally prove that the optimal axis-aligned predictor constitutes a rigorous upper bound on the minimum training Mean Squared Error (MSE) attainable by any linear or kernel-based regressor defined on the same quantum feature map. Since computing this exact bound requires an intractable scan of the full Pauli basis, we introduce a Monte Carlo framework to efficiently estimate it using a tractable subset of measurement axes. We further provide non-asymptotic statistical guarantees to certify performance within a practical measurement budget. This method enables rapid comparison of quantum feature maps and early diagnosis of expressivity, allowing for the informed selection of architectures before deploying higher-complexity models.
Paper Structure (16 sections, 4 theorems, 23 equations, 3 tables, 1 algorithm)

This paper contains 16 sections, 4 theorems, 23 equations, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Pauli-axis Upper Bound for Quantum Regression
  3. Axis-Aligned Least-Squares Upper Bound
  4. Analytical Derivation and Complexity
  5. Monte Carlo Estimation Framework
  6. Statistical Guarantees and Threshold Selection
  7. Adaptive Sample-Size Calibration
  8. Experimental Setup
  9. Datasets and Benchmarking Strategy
  10. Quantum Feature Map and Configuration Space
  11. Numerical Performance and Benchmarking
  12. Results and Discussion
  13. 1. Real-World Complexity and Linear Combinations.
  14. 2. Feature Map Alignment in the Sparse Regime.
  15. 3. The "Dense Trap" in Correlated Data.
  16. ...and 1 more sections

Key Result

Theorem 1

For any dataset $D$ encoded by a fixed quantum feature map, the optimal affine linear regression error is upper bounded by the best axis-aligned error:

Theorems & Definitions (8)

  • Theorem 1: Pauli-axis upper bound
  • proof
  • Theorem 2: Monte Carlo bound
  • proof
  • Theorem 3
  • proof
  • Corollary 4
  • proof