Quantum Measurement Statistics as Bayesian Uncertainty Estimators for Physics-Constrained Learning

TL;DR

This paper links quantum measurement statistics from variational quantum circuits to Bayesian uncertainty, demonstrating a natural, efficient way to quantify uncertainty in physics-informed machine learning without complex Bayesian neural networks.

Contribution

It establishes a formal connection between quantum measurement statistics and Bayesian uncertainty, enabling calibrated UQ in physics-constrained models with fewer computational resources.

Findings

Quantum UQ achieves near-target coverage probabilities with N >= 5000 shots.

Physics-constrained circuits significantly reduce calibration error compared to unconstrained ones.

Quantum circuits extract more UQ information per evaluation than classical methods.

Abstract

Uncertainty quantification (UQ) is essential for deploying machine learning models in safety-critical physical systems, yet classical Bayesian approaches incur substantial computational overhead. We establish a formal connection between Born-rule measurement statistics from variational quantum circuits (VQCs) and Bayesian posterior uncertainty, proving that repeated quantum measurements naturally produce calibrated prediction intervals without requiring explicit Bayesian neural network (BNN) machinery. We demonstrate this framework on physics-constrained VQCs trained on PDE residuals. Systematic experiments comparing quantum shot-based UQ against MC Dropout and Deep Ensemble baselines show that quantum UQ achieves coverage probabilities within 1-3% of target confidence levels at N >= 5000 shots, while MC Dropout systematically over-covers by 4-5%. Physics-constrained circuits reduce the expected calibration error (ECE) by 34-40% compared to unconstrained counterparts, with interval widths 14-30% narrower at equivalent coverage. Information-theoretic analysis reveals that quantum circuits extract ~15% more bits of UQ information per evaluation than MC Dropout and ~42% more than Deep Ensembles (M = 10), owing to the exponential Hilbert space accessible through Born-rule sampling. These results establish quantum measurement statistics as a principled, computationally efficient framework for uncertainty quantification in physics-informed learning.