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

TL;DR

This paper introduces a training-free, certified error bound for quantum regression based on Pauli expectation values, enabling efficient performance estimation and model selection without extensive training.

Contribution

It provides a novel, rigorous upper bound on quantum regression error derived from Pauli measurements, with a scalable Monte Carlo estimation method and statistical guarantees.

Findings

The bound accurately predicts regression error in quantum models.

The Monte Carlo method efficiently estimates the bound with limited measurements.

The approach aids in selecting quantum architectures before complex training.

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.