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arXiv:2511.12613

Quantum Orthogonal Separable Physics-Informed Neural Networks

Pietro Zanotta, Ljubomir Budinski, Caglar Aytekin, Valtteri Lahtinen

quant-phphysics.comp-phSubmitted Nov 16, 2025• Announced Nov 18, 2025• 0 citations• Revised Oct 28, 2026 (v2)
v1 · Nov 16, 2025v2 · Oct 28, 2026
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Abstract

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 $\mathcal O(d\log d/ε^2)$, which is asymptotically smaller than the $Θ(d^2)$ cost of conventional direct dense multiplication when $ε(d)^{-2}\log d=o(d)$. 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.

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Quantum physics-informed neural networks (QPINNs) have recently emerged as a promising framework for the solution of partial differential equations (PDEs), with several studies reporting improved convergence and accuracy relative to classical physics-informed neural networks (PINNs) at reduced training cost. Motivated by these advances, we introduce geometric quantum physics-informed neural networks (GQPINNs), a symmetry-aware extension of QPINNs in which the geometric structure of the underlying PDE is incorporated directly into the quantum-circuit ansatz. Building on the framework of geometric quantum machine learning, we construct parametrized circuits that encode finite-group and compact Lie-group symmetries as inductive biases through problem-specific equivariant generator sets . Using a twirling-based construction, we derive symmetry-preserving gates that ensure that the model predictions respect the symmetries of the governing equation whenever the boundary and initial data are symmetry compatible. We benchmark GQPINNs against standard QPINNs and symmetry-adapted classical PINN baselines under matched training protocols across a representative set of linear and nonlinear PDEs. Across these benchmarks, GQPINNs achieve improved solution accuracy, as quantified by lower mean absolute error, while requiring substantially fewer trainable parameters. These results identify symmetry-aware quantum-circuit design as an effective route toward improved efficiency and generalization in quantum PDE solvers and provide a systematic framework for incorporating geometric inductive biases into quantum-enhanced scientific machine learning.
Physics-informed neural networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding governing physical laws directly into the training objective. Recent advances in quantum machine learning have motivated hybrid quantum-classical extensions aimed at enhancing representational capacity while remaining compatible with near-term quantum hardware. In this work, we investigate trainable embedding strategies within quantum-assisted PINNs for solving parabolic PDEs, using one- and two-dimensional heat equations as canonical benchmarks. We introduce two quantum-assisted architectures that differ in their embedding components. In the first approach, a classical feed-forward neural network generates trainable feature maps for quantum data encoding (FNN-TE-QPINN). In the second, the embedding stage is realized entirely by a parameterized quantum circuit (QNN-TE-QPINN), yielding a fully quantum feature map. Our findings emphasize the critical role of embedding design and support hybrid quantum-classical approaches for parabolic PDE modeling in the NISQ era.
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.
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.
Neural Physics recasts local discretisations of partial differential equations (PDEs) as fixed convolutional operators, providing a physics-preserving alternative to data-driven surrogate modelling in scientific machine learning. However, existing realizations remain largely confined to classical AI hardware and do not directly connect to quantum structured operator design. To bridge this gap, we introduce a \emph{Quantum Neural Physics} framework and develop a Hybrid Quantum-Classical CNN Multigrid Solver (HQC-CNNMG). The proposed method maps analytically prescribed stencil operators to local quantum convolutional primitives and embeds them within a classical multilevel W-cycle architecture, combining the operator-centric view of scientific ML with the numerical rigor of multigrid solvers. Using amplitude encoding together with the Linear Combination of Unitaries (LCU) and the Quantum Fourier Transform (QFT), the resulting local quantum operators admit logarithmic-depth implementation, with circuit depth scaling as O(log⁡K)\mathcal{O}(\log K)O(logK) for an encoded block of size KKK under the idealized parallel circuit model considered here. Numerical experiments on Poisson, transient diffusion, convection--diffusion, and incompressible Navier--Stokes problems demonstrate numerical consistency, stable multilevel behaviour, and workflow-level feasibility on noiseless simulators. Comparisons with representative quantum linear solver paradigms further show that the main strength of HQC-CNNMG lies in its balanced trade-off among local circuit depth, numerical robustness, and compatibility with PDE structure, rather than in fully quantum global inversion.