Physics-Informed Classical-Quantum ML

• The framework is a hybrid model that embeds physical laws such as differential equations and conservation laws into classical-quantum architectures to ensure physics-consistent predictions.
• It leverages variational quantum circuits alongside classical neural networks to achieve significant parameter reduction and training acceleration in solving complex problems.
• Its design effectively addresses forward and inverse tasks—including PDEs, Hamiltonian learning, and quantum control—offering enhanced robustness and computational efficiency.

A Physics-Informed Classical-Quantum Machine Learning Framework systematically integrates domain knowledge from physics into hybrid architectures involving both classical machine learning and quantum algorithms. These frameworks are distinguished by their explicit embedding of physical models—such as differential equations, conservation laws, and operator constraints—into the structure, loss function, or data processing loop of quantum-classical networks. The result is a modeling paradigm that leverages quantum resources for expressivity or computational acceleration, while ensuring that predictions remain consistent with known physical laws.

1. Formalism and Problem Scope

Physics-informed classical-quantum machine learning frameworks address both forward and inverse problems across classical and quantum domains, including:

• Solving ordinary/partial/integro-differential equations (ODEs, PDEs, IDEs) with boundary and initial conditions.
• Quantum system characterization, Hamiltonian learning, and control optimization.
• Efficient subspace discovery in quantum simulation or electronic structure.

The fundamental structure involves a hybrid model, typically combining parameterized quantum circuits (with variational parameters θ) and classical neural architectures (with weights w), with the overall objective of minimizing a composite loss function. This loss contains terms enforcing both data fidelity and satisfaction of operator constraints derived from the governing equations of the physical system.

A generic total loss in these frameworks can be expressed as: $$\mathcal{L}_{\rm total} = \mathcal{L}_{\rm data} + \lambda_{\rm phys} \mathcal{L}_{\rm phys} + \lambda_{\rm bc} \mathcal{L}_{\rm bc} + \lambda_{\rm reg} \mathcal{L}_{\rm reg}$$ where $\mathcal{L}_{\rm phys}$ encodes physics constraints (e.g., differential or operator residuals) and the other terms incorporate experimental data, boundary/initial conditions, or regularization (Farea et al., 20 Mar 2025, Klement et al., 21 Jan 2026, Koutromanos et al., 2024, Dehaghani et al., 2024, Setty et al., 2023, Lantigua et al., 10 Nov 2025).

2. Quantum-Classical Model Architectures

Central to these frameworks is the flexible coupling of classical neural networks and quantum circuits:

• Serial and Parallel Integration: Quantum circuits may serve as surrogate networks for solution functions (e.g., replacing deep neural networks in PINNs), be embedded inside classical pipelines, or interact multiplicatively/additively with classical predictors, as in QPINN-MAC (Lantigua et al., 10 Nov 2025).
• Variational Quantum Circuits: Parameterized quantum circuits provide trial solutions whose outputs, when measured, represent state variables, observables, or latent encodings depending on the problem domain (Markidis, 2022, Setty et al., 2023).
• Hybrid Layering and Data Flow: Some architectures interleave classical preprocessing layers, quantum PQCs, and classical postprocessing layers around physics-informed loss calculations (Farea et al., 20 Mar 2025, Koutromanos et al., 2024).

Common Quantum Ansätze

• Discrete-variable (DV) circuits: e.g., data reuploading circuits or hardware-efficient layers with entanglers (Klement et al., 21 Jan 2026).
• Continuous-variable (CV) circuits: e.g., photonic networks using displacement, squeezing, Kerr gates (Markidis, 2022, Dehaghani et al., 2024).
• Quantum Orthogonal layers: leveraging quantum algorithms for matrix-vector multiplication within neural layers, e.g., QO-MLP (Zanotta et al., 16 Nov 2025).
• Latent-space encoding: basis states map to component functions or features, with products and derivatives constructed by quantum arithmetic (Paine et al., 2023).

def forward(model, x): # x: input variables h_classical = classical_preprocessor(x) q_result = quantum_circuit(h_classical) u_pred = classical_postprocessor(q_result) return u_pred

This structure enables backpropagation of gradients through both quantum and classical components, with quantum gradients typically computed via parameter-shift rules or finite differences.

3. Physics-Informed Loss Construction

The defining feature of these frameworks is loss functions that penalize violation of physical laws. Methodologies include:

• Operator Residuals: E.g., for a PDE $\mathcal{D}[u_\theta]=0$, the loss enforces $\| \mathcal{D}[u_\theta] \|^2_{L^2}$ on collocation points (Farea et al., 20 Mar 2025, Klement et al., 21 Jan 2026).
• Boundary/Initial/Integral Constraints: Terms enforcing Dirichlet, Neumann, or more general conditions, along with on-the-fly calculation of required integral transforms or moments (Kumar et al., 2022, Setty et al., 2023).
• Automatic/Quantum Differentiation and Integration: Derivatives and integrals with respect to inputs or parameters are evaluated by autodiff (classical) or parameter-shift/analytic methods (quantum circuits) (Setty et al., 2023, Markidis, 2022, Paine et al., 2023).
• Composite Losses for Optimal Control or Inverse Problems: For quantum control, loss incorporates PMP or Lindblad equations, in addition to classical regularization (Dehaghani et al., 2024, Liu et al., 12 Jun 2025).

A salient example: $$\mathcal{L}(\theta) = \lambda_{res} \| \mathcal{N}[u_\theta](\mathbf{x}) \|^2 + \lambda_{bc} \| u_\theta|_{\Gamma} - g \|^2 + \lambda_{ic} \| u_\theta|_{\Omega_0} - h \|^2$$ (Farea et al., 20 Mar 2025, Klement et al., 21 Jan 2026)

4. Computational Methods and Training Protocols

Optimization Algorithms

• Stochastic Gradient Descent (SGD): Robust for quantum PINNs, more effective than adaptive optimizers for small circuits suffering from landscape traps (Markidis, 2022).
• Parameter-Shift Rule: Used for accurate, low-variance gradient evaluation of quantum circuit parameters.
• Adaptive Scheduling: Hybrid frameworks often employ lr-scheduling and mini-batch strategies. For quantum-classical parameter partitioning, both gradients are accumulated and updated within unified optimizers (Farea et al., 20 Mar 2025, Koutromanos et al., 2024).

Computational Acceleration

• GPU/Parallel Support: Native GPU acceleration when all operations are built on differentiable tensor libraries such as PyTorch (Koutromanos et al., 2024).
• Quantum Speedup Mechanisms: Fast matrix-vector multiplication (O(d log d/ε²) vs classical O(d²)), efficient handling of large latent spaces, and global constraint enforcement via state overlaps (Zanotta et al., 16 Nov 2025, Paine et al., 2023).

Software Examples

• TorchQC: Embeds all quantum operations as PyTorch tensor computations, enabling seamless integration with modern ML pipelines (Koutromanos et al., 2024).

5. Numerical Performance and Comparative Metrics

Hybrid physics-informed classical-quantum frameworks consistently demonstrate:

• Parameter Efficiency: Hybrid networks matching or surpassing accuracy of pure classical PINNs with an order-of-magnitude fewer parameters (Farea et al., 20 Mar 2025).
• Training Acceleration: Quantum-enhanced PINNs (qPINNs) reach prescribed error thresholds in 10–100× fewer epochs on benchmark PDEs (Klement et al., 21 Jan 2026).
• Stability and Extrapolation: Physics-regularized losses yield sharper convergence, lower error, and robustness to measurement noise compared to data-only learning (Liu et al., 12 Jun 2025, Setty et al., 2023).

Representative metrics from (Farea et al., 20 Mar 2025, Klement et al., 21 Jan 2026):

Framework	Epochs to MSE < 10⁻⁴	Relative L₂ Error	Parametric Reduction
Classical PINN	100,000	20.3%	1.0×
QCPINN	1,000	3.3%	~0.1×

Limitations observed include the onset of barren plateaus in quantum landscapes, higher shot requirements for circuit-based differentiation, and the challenge of scaling to high-dimensional PDEs or large system sizes (Klement et al., 21 Jan 2026, Markidis, 2022, Lantigua et al., 10 Nov 2025).

6. Applications and Specialized Extensions

Quantum Control and Dynamics

Hybrid frameworks are deployed for quantum optimal control, enabling state-to-state transfer with state fidelities exceeding 0.99, using a PINN structure to optimize discretized control fields under physics constraints (PMP, Lindblad) (Dehaghani et al., 2024).

Hamiltonian Learning

Inverse PINN-HL frameworks jointly optimize NNQS representations and Hamiltonian parameters, enforcing the Schrödinger equation as a physics constraint. MSE scaling approaches the Heisenberg limit, outperforming purely data-driven tomography in both accuracy and resource efficiency (Liu et al., 12 Jun 2025).

PDEs, UQ, and Operator Theory

In high-dimensional PDEs, QO-SPINN exploits quantum-orthogonal building blocks to accelerate matrix-vector operations, enabling robust uncertainty quantification through principled GP priors over Lipschitz-constrained solution spaces (Zanotta et al., 16 Nov 2025).

Quantum Sampling and Combinatorial Screening

In sample-based quantum diagonalization workflows, physics-informed pruning via implicit perturbative screening guides generative models (e.g., RBM), compressing the relevant Hilbert subspace and improving energy convergence in noisy quantum simulations (Patra et al., 7 Dec 2025).

7. Theoretical Guarantees and Future Directions

• Universal Approximation: Formal proofs exist for hybrid architectures (e.g., QPINN-MAC) which combine classical and quantum components, showing universal function-approximation power with error converging as $O((N\mathcal N)^{-1/2})$ (Lantigua et al., 10 Nov 2025).
• Barren Plateau Mitigation: Strategic quantum-classical coupling prevents gradient collapse, ensuring that gradient norms decay only polynomially with system size, not exponentially, preserving trainability in deep circuits (Lantigua et al., 10 Nov 2025).
• Scalability: Prospects include extension to many-body quantum simulation, reinforcement learning environments, integral/differential operator learning, and large-scale, hardware-integrated workflows (Koutromanos et al., 2024, Patra et al., 7 Dec 2025).

Research on loss landscape geometry, advanced ansatz design (e.g., quantum convolutional blocks, skip connections), and rigorous error/uncertainty quantification remains ongoing, with applications expanding from quantum control to quantum chemistry and large-scale dynamical modeling.

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Overall, Physics-Informed Classical-Quantum Machine Learning frameworks synthesize the rigor of physics-based modeling with the representational power and trainability of hybrid quantum-classical architectures, yielding significant advances in accuracy, efficiency, and interpretability across simulation, control, and inverse problems in both the quantum and classical domains (Koutromanos et al., 2024, Markidis, 2022, Farea et al., 20 Mar 2025, Zanotta et al., 16 Nov 2025, Liu et al., 12 Jun 2025, Klement et al., 21 Jan 2026).