Abstract Operator learning enables fast surrogate modelling of high-dimensional dynam-ical systems, but existing approaches face two fundamental limitations: the quadratic cost of dense neural layers and unreliable uncertainty quantification in safety-critical settings. We propose Conformalized Quantum Deep Operator Network (DeepONet) Ensembles, a framework that addresses both challenges simultaneously. Using Quantum Orthogonal Neural Networks (QOrthoNNs), we characterize the resource regime in which their established˜︁established˜︁ O(n) hidden-layer running-time scaling improves on the O(n 2) cost of a classical dense layer. To quantify uncertainty, we combine ensemble predictions with split conformal calibration. For a new input–output function pair jointly exchangeable with the calibration pairs, we prove a finite-sample, distribution-free lower bound on the expected fraction of covered query locations. As a secondary proof-of-concept, we explore hybrid classical–quantum architectures and superposed execution to manage ensemble runtime and hardware requirements. Experiments on synthetic operator benchmarks and real-world power-system dynamics show accurate predictions under ideal simulation and empirical coverage near the target under both ideal conditions and selected compact-circuit simulations with depolarizing 1 or device-calibrated composite noise. Together, these results connect resource-aware scalability analysis with finite-sample uncertainty guarantees for quantum operator learning.
Abstract Operator learning enables fast surrogate modelling of high-dimensional dynam-ical systems, but existing approaches face two fundamental limitations: the quadratic cost of dense neural layers and unreliable uncertainty quantification in safety-critical settings. We propose Conformalized Quantum Deep Operator Network (DeepONet) Ensembles, a framework that addresses both challenges simultaneously. Using Quantum Orthogonal Neural Networks (QOrthoNNs), we characterize the resource regime in which their established˜︁established˜︁ O(n) hidden-layer running-time scaling improves on the O(n 2) cost of a classical dense layer. To quantify uncertainty, we combine ensemble predictions with split conformal calibration. For a new input–output function pair jointly exchangeable with the calibration pairs, we prove a finite-sample, distribution-free lower bound on the expected fraction of covered query locations. As a secondary proof-of-concept, we explore hybrid classical–quantum architectures and superposed execution to manage ensemble runtime and hardware requirements. Experiments on synthetic operator benchmarks and real-world power-system dynamics show accurate predictions under ideal simulation and empirical coverage near the target under both ideal conditions and selected compact-circuit simulations with depolarizing 1 or device-calibrated composite noise. Together, these results connect resource-aware scalability analysis with finite-sample uncertainty guarantees for quantum operator learning.
In the realm of computational science and engineering, constructing models that reflect real-world phenomena requires solving partial differential equations (PDEs) with different conditions. Recent advancements in neural operators, such as deep operator network (DeepONet), which learn mappings between infinite-dimensional function spaces, promise efficient computation of PDE solutions for a new condition in a single forward pass. However, classical DeepONet entails quadratic complexity concerning input dimensions during evaluation. Given the progress in quantum algorithms and hardware, here we propose to utilize quantum computing to accelerate DeepONet evaluations, yielding complexity that is linear in input dimensions. Our proposed quantum DeepONet integrates unary encoding and orthogonal quantum layers. We benchmark our quantum DeepONet using a variety of PDEs, including the antiderivative operator, advection equation, and Burgers' equation. We demonstrate the method's efficacy in both ideal and noisy conditions. Furthermore, we show that our quantum DeepONet can also be informed by physics, minimizing its reliance on extensive data collection. Quantum DeepONet will be particularly advantageous in applications in outer loop problems which require exploring parameter space and solving the corresponding PDEs, such as uncertainty quantification and optimal experimental design.
General operator learning for parametric partial differential equations (PDEs) is a fundamental challenge at the intersection of artificial intelligence and physics-based modeling. Physics-informed Deep Operator Networks (PI-DeepONets) incorporate governing equations into learning, but face optimization difficulties in high dimensions. Quantum extensions offer additional representational capacity, yet their trainability can be hindered by barren plateaus. Here, we introduce QPI-DeepONet-MAC, a hybrid quantum-classical architecture that integrates parameterized quantum circuits into PI-DeepONets through a multiplicative-and-additive coupling (MAC) of classical representations and quantum expectation values. We prove analytically that the proposed architecture retains universal operator approximation capabilities. Furthermore, we derive bounds on quantum parameter gradients, establishing explicit conditions to avoid barren plateaus alongside a scaling criterion for trainability. Additionally, we derive a lower bound on the number of sensors required for a target accuracy, directly linking sampling requirements to input function regularity, domain dimension, and PDE dynamical growth. Finally, we propose an informational regularization scheme based on quantum coherence and state fidelity dynamics to prevent coherence loss and preserve quantum structure during training. Together, these results establish a theoretical foundation for quantum-enhanced physics-informed operator learning and position QPI-DeepONet-MAC as a scalable, stable framework for solving parametric PDEs.
We introduce Quantum Orthogonal Separable Physics-Informed Neural Networks (QO-SPINNs), which replace selected dense hidden transformations in physics-informed PDE models with quantum-compatible orthogonal maps. A pyramidal reconfigurable-beamsplitter circuit realizes these maps with Θ(d) circuit depth. Under the stated signed-readout access model and fixed per-call failure probability, a width-d map has sequential access-cost upper bound O(dlogd/ε2), which is asymptotically smaller than the Θ(d2) cost of conventional direct dense multiplication when ε(d)−2logd=o(d). Because unary state preparation discards magnitude, we retain each hidden-state norm classically and restore it after readout, yielding the exact map Wh under ideal readout. We study this construction in two complementary settings: forward PDE regression and a norm-controlled residual model for representation geometry and uncertainty quantification. In the norm-controlled model, structural orthogonality yields explicit bi-Lipschitz bounds, while finite-readout analysis relates sufficient reconstruction precision to a prescribed physical-coordinate resolution through trained-model constants. This makes the access-cost condition assessable for trained checkpoints, although favorable fixed-resolution scaling across widths is not established. A post-hoc Bayesian-linear covariance provides uncertainty from the same frozen predictor, and cross-PDE transfer shows that strong distance-awareness can persist even when absolute coverage calibration fails. All Quantum Orthogonal (QO) layers are evaluated by classical simulation: the results therefore characterize a quantum-compatible structured neural primitive rather than demonstrating a practical quantum-hardware advantage.
Quantum computing introduces unique forms of randomness arising from measurement processes, gate noise, and hardware imperfections. Ensuring reliable uncertainty quantification in such quantum-driven or quantum-derived predictions is an emerging challenge. In classical machine learning, conformal prediction has proven to be a robust framework for distribution-free uncertainty calibration, often focusing on univariate or low-dimensional outputs. Recent advances (e.g., [1–3]) have extended conformal methods to handle multi-output or multi-dimensional responses, addressing sophisticated tasks such as time-series, image classification sets, and quantum-generated probability distributions. However, bridging the gap between these powerful conformal frameworks and the high-dimensional, noise-prone distributions typical of quantum measurement scenarios remains largely open. In this paper, we propose a unified approach to harness quantum conformal methods for multi-output distributions, with a particular emphasis on two experimental paradigms: (i) a standard 2-qubit circuit scenario producing a four-dimensional outcome distribution, and (ii) a multi-basis measurement setting that concatenates measurement probabilities in different bases (Z, X, Y) into a twelve-dimensional output space. By combining a multi-output regression model (e.g., random forests) with distributional conformal prediction, we validate coverage and interval-set sizes on both simulated quantum data and multi-basis measurement data. Our results confirm that classical conformal prediction can effectively provide coverage guarantees even when the target probabilities derive from inherently quantum processes. Such synergy opens the door to next-generation quantum-classical hybrid frameworks, providing both improved interpretability and rigorous coverage for quantum machine learning tasks. All codes and full reproducible Colab notebooks are made available at https://github.com/detasar/QECMMOU.
Quantum machine learning seeks to leverage quantum computers to improve upon classical machine learning algorithms. Currently, robust uncertainty quantification methods remain underdeveloped in the quantum domain, despite the critical need for reliable and trustworthy predictions. Recent work has introduced quantum conformal prediction, a framework that produces prediction sets that are guaranteed to contain the true outcome with a user-specified probability. In this work, we formalise how the time-varying noise inherent in quantum processors can undermine conformal guarantees, even when calibration and test data are exchangeable. To address this challenge, we draw on Adaptive Conformal Inference, a method which maintains validity over time via repeated recalibration. We introduce Adaptive Quantum Conformal Prediction (AQCP), an algorithm which provides asymptotic average coverage guarantees under arbitrary hardware noise conditions. Empirical studies on an IBM quantum processor demonstrate that AQCP achieves the target coverage level and exhibits greater stability than quantum conformal prediction.
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.