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arXiv · 2601.00745

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

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

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Demerson N. Gonçalves, Tharso D. Fernandes, Pedro H. G. Lugao, João T. Dias. 2026-01-02. Training-Free Certified Bounds for Quantum Regression: A Scalable Framework. https://arxiv.org/abs/2601.00745

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