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.
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14 pages, 6 figures, 5 tables. Code available at https://github.com/nimanpra/quantum_state_uq
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.
To make predictions on the future states of engineering systems, prognostics has been increasingly relying on data-driven models and machine learning. Recent work has also looked at the potential of quantum machine learning to provide insight on the health state of systems. Nevertheless, these approaches treated the quantum component only as a deterministic predictor, in fact disregarding the uncertainty information. In this work, we propose a different approach to exploiting quantum circuits in prognostics. We focus on Bayesian learning of neural networks for Remaining Useful Life (RUL) prediction and uncertainty quantification. Here, the quantum circuit is introduced not as a data-driven model, but as a generator of neural network weights. Using Variational Inference, the quantum circuit can be trained to approximate the true posterior of the classical machine learning predictor. The overall method retains the data-processing ability of state-of-the-art machine learning models, while exploiting the quantum circuit to introduce uncertainty by sampling the modelspace from a distribution that is classically nontrivial. We validate our approach on the task of predicting the End of Discharge of Li-ion batteries, using data generated from a simulator with tunable process uncertainty, and we compare the predictions obtained through quantum sampling with those from Flipout Bayesian neural networks, heteroscedastic neural networks and Monte Carlo Dropout. The results show that our quantum circuits learn to approximate the weight posterior and that the resulting data-driven models demonstrate accuracy and uncertainty quantification that is comparable if not superior to the baselines. Overall, our work demonstrates the potential of quantum computing for uncertainty-aware prognostics, and sets the stage for further investigations in this area.
Quantum fidelity estimation is a fundamental task in quantum information and computing to benchmark the quality of quantum states, and hence of quantum processes, on noisy quantum hardware. Point estimates of fidelity by existing methods generally lack the rigorous uncertainty quantification required for reliable preparation and verification of these tasks. While Bayesian and bootstrap frameworks are widely used for uncertainty quantification, they are sensitive to model misspecification or lack formal coverage guarantees in non-asymptotic regimes. This paper introduces a hybrid "Bayesian-informed" conformal prediction-based framework to provide distribution-free, rigorous prediction intervals for quantum state fidelity estimation which can be used, with care, to approximate confidence intervals with respect to the quantum randomness of Born's rule. The framework requires a point estimator, and we consider both classical point estimators as well as data-driven neural net estimators. We evaluate the robustness of the methodology across a diverse range of experimental challenges, including finite-sample noise, tomographic incompleteness, physical model error, and out-of-distribution data. Our results demonstrate that conformalized estimators maintain statistical validity where traditional methods fail, offering a scalable and reliable approach for the verification of modern quantum systems. The best performing variant of our methodology is based on a class-conditional conformal prediction, as this most closely approximates the conditional coverage desired for a true confidence interval.
We propose a Hybrid Quantum-Classical Physics-Informed Neural Network (HQC-PINN) that integrates parameterized variational quantum circuits into the PINN framework for hydrological PDE-constrained learning. Our architecture encodes multi-source remote sensing features into quantum states via trainable angle encoding, processes them through a hardware-efficient variational ansatz with entangling layers, and constrains the output using the Saint-Venant shallow water equations and Manning's flow equation as differentiable physics loss terms. The inherent stochasticity of quantum measurement provides a natural mechanism for uncertainty quantification without requiring explicit Bayesian inference machinery. We further introduce a quantum transfer learning protocol that pre-trains on multi-hazard disaster data before fine-tuning on flood-specific events. Numerical simulations on multi-modal satellite and meteorological data from the Kalu River basin, Sri Lanka, show that the HQC-PINN achieves convergence in ~3x fewer training epochs and uses ~44% fewer trainable parameters compared to an equivalent classical PINN, while maintaining competitive classification accuracy. Theoretical analysis indicates that hydrological physics constraints narrow the effective optimization landscape, providing a natural mitigation against barren plateaus in variational quantum circuits. This work establishes the first application of quantum-enhanced physics-informed learning to hydrological prediction and demonstrates a viable path toward quantum advantage in environmental science.
Quantum circuit design is a key bottleneck for practical quantum machine learning on complex, real-world data. We present an automated framework that discovers and refines variational quantum circuits (VQCs) using graph-based Bayesian optimization with a graph neural network (GNN) surrogate. Circuits are represented as graphs and mutated and selected via an expected improvement acquisition function informed by surrogate uncertainty with Monte Carlo dropout. Candidate circuits are evaluated with a hybrid quantum-classical variational classifier on the next generation firewall telemetry and network internet of things (NF-ToN-IoT-V2) cybersecurity dataset, after feature selection and scaling for quantum embedding. We benchmark our pipeline against an MLP-based surrogate, random search, and greedy GNN selection. The GNN-guided optimizer consistently finds circuits with lower complexity and competitive or superior classification accuracy compared to all baselines. Robustness is assessed via a noise study across standard quantum noise channels, including amplitude damping, phase damping, thermal relaxation, depolarizing, and readout bit flip noise. The implementation is fully reproducible, with time benchmarking and export of best found circuits, providing a scalable and interpretable route to automated quantum circuit discovery.
Abstract We show that ensembles of deep neural networks, called deep ensembles, can be used to perform quantum parameter estimation while also providing a means for quantifying uncertainty in parameter estimates, which is a key advantage of using Bayesian inference for parameter estimation that is lost when using existing machine learning methods. We show that optimizing for both accurate parameter estimates and well calibrated uncertainty estimates does not lead to degradation in the former as opposed to only optimizing for accuracy. We also show that the drift detection capabilities of these ensemble models can be used to detect drift in the experimental data used during inference. This approach is also shown to provide much faster inference time than both likelihood-based and likelihood-free Bayesian inference. These results suggest that such models could enable accurate, real-time parameter estimation with quantified uncertainty, making them promising candidates for deployment in experimental settings.
Quantum machine learning (QML) models introduce unique sources of uncertainty arising from finite measurement shots, hardware noise, and variational optimization instability. Despite rapid progress in quantum algorithms, systematic uncertainty quantification (UQ) in QML remains largely underexplored and lacks standardized tooling. We present Quan-tumUQ, an open-source Python library for uncertainty quantification in quantum machine learning with native support for PennyLane and Qiskit. QuantumUQ provides shot-based uncertainty estimation, ensemble-based epistemic uncertainty, measurement-budget stability profiling, and calibration metrics for classification and regression tasks. We formalize the uncertainty decomposition in QML, describe the architectural design of the library, and demonstrate its capabilities through reproducible experiments. Our results show that measurement-induced uncertainty exhibits predictable scaling behavior and that calibration analysis reveals systematic overconfidence in variational quantum classifiers. QuantumUQ aims to make robustness, reliability, and trustworthiness first-class citizens in quantum ML research.