Hybrid Quantum-Classical PINNs

Updated 4 July 2026

HQPINNs are physics-informed models where classical neural networks incorporate quantum subroutines for feature mapping and latent transformations.
They employ diverse architectures such as fixed quantum feature maps, latent-space quantum processors, and trainable-embedding circuits to enforce PDE constraints.
Empirical studies indicate improved convergence and reduced errors in smooth, stiff, or multiscale PDEs, though challenges persist with discontinuities and hardware noise.

Searching arXiv for recent HQPINN and related QPINN papers to ground the article. Hybrid Quantum-Classical Physics-Informed Neural Network (HQPINN) denotes a class of physics-informed learning architectures in which a physics-informed neural network is coupled to a quantum component, typically a parameterized quantum circuit or a quantum feature map, while residual construction, loss assembly, and optimization remain partly or predominantly classical. Across the literature, the term covers several distinct patterns: fixed quantum feature augmentation for classical PINNs, late-stage quantum latent processors embedded inside otherwise classical PINNs, quantum surrogate models trained through classical physics-informed losses, and trainable-embedding quantum-assisted PINNs for time-dependent PDEs. What unifies these variants is that the learned approximation is constrained by governing equations rather than labeled solution data alone, and that a quantum subroutine contributes either representational structure, latent transformation, or trainable nonlinear features within a hybrid differentiable workflow (Soni, 17 Sep 2025, Sedykh et al., 2023, Tran et al., 16 Feb 2026).

1. Conceptual scope and defining characteristics

HQPINNs are most accurately understood as a family rather than a single architecture. In the minimal form, a classical PINN is augmented with quantum-circuit-generated features and trained using an ordinary residual loss. A proof-of-concept example solves the initial value problem

$\frac{dy}{dx}+2y=0,\qquad y(0)=1,$

by sending each scalar collocation point $x\in[0,1]$ through a two-qubit circuit, measuring computational-basis probabilities, concatenating the resulting four-dimensional quantum feature vector with the original input, and learning the solution with a classical feed-forward network under a hard initial-condition ansatz (Soni, 17 Sep 2025). In that setting, the quantum part is fixed feature extraction rather than a trainable variational layer.

A stronger notion of hybridization inserts a trainable quantum circuit into the interior of the neural approximator. In a 3D computational fluid dynamics application, a classical multilayer perceptron maps spatial coordinates $(x,y,z)$ to a 16-dimensional latent representation, then a parallel hybrid head routes 15 units to a variational quantum circuit and one unit to a classical branch before a final dense layer predicts $(u,v,w,p)$ for the steady incompressible Navier–Stokes equations in a parameterized Y-shaped mixer (Sedykh et al., 2023). In this design, the PINN remains classical in its loss construction and PDE differentiation, while the latent representation is partly processed quantumly.

Another line replaces the usual classical surrogate altogether with a quantum model while retaining a classical physics-informed training loop. Continuous-variable QPINNs for Poisson and heat equations use qumode circuits as the trainable function approximator and compute PDE residuals classically through autodiff or consistency-based derivative constructions (Markidis, 2022, Panichi et al., 15 Mar 2025). A plausible implication is that the boundary between “HQPINN” and “QPINN” is architectural rather than categorical: some works use the quantum model as one module inside a classical network, whereas others use the quantum model as the surrogate and preserve hybridization at the training-loop level.

More recent work frames HQPINNs explicitly as hybrid classical-to-quantum-to-classical solvers. For nonlinear PDEs such as Burgers, Allen–Cahn, and Korteweg–de Vries, a classical backbone first extracts latent features from $(x,t)$ , a parameterized quantum circuit transforms a bounded latent vector through angle encoding and entangling layers, and a final classical dense layer maps quantum measurements to the scalar field $u(x,t)$ (Zabihi et al., 3 Jun 2026). Closely related trainable-embedding quantum-assisted PINNs for parabolic and reaction-diffusion PDEs study whether the embedding from coordinates to quantum angles should be classical or quantum, identifying the embedding stage itself as a primary design variable rather than mere preprocessing (Tran et al., 16 Feb 2026, Tran et al., 10 Feb 2026).

2. Architectural patterns

The literature supports several recurring HQPINN motifs.

The first is the quantum feature map plus classical PINN pattern. In the ODE proof of concept, the trial solution is

$\hat y(x)=1+x\,g_\theta([x,q(x)]),$

where $g_\theta$ is a classical MLP and $q(x)$ is a quantum feature vector obtained from a two-qubit circuit

$U(t)=\big(Rx(-2t)\otimes Rx(-2t)\big)(H\otimes I),\qquad t=x.$

The measurement output is

$x\in[0,1]$ 0

and the fused input is $x\in[0,1]$ 1 (Soni, 17 Sep 2025). The initial condition is enforced exactly by construction.

The second is the late latent-space quantum processor. In the Y-mixer CFD architecture, the classical front-end has 5 hidden layers of 64 neurons, followed by a 16-dimensional penultimate layer. The hybrid head splits those 16 units into 15 quantum inputs and 1 classical scalar. The quantum branch uses repeated encoding blocks with depth $x\in[0,1]$ 2, followed by variational layers repeated $x\in[0,1]$ 3, and returns expectation values such as

$x\in[0,1]$ 4

that are fused with the classical branch before a final dense map to $x\in[0,1]$ 5 (Sedykh et al., 2023). This architecture treats the PQC as a latent feature transformer rather than as the entire solution network.

The third is the classical–quantum–classical serial architecture. Work on explaining the advantage of quantum-enhanced PINNs embeds a three-qubit variational circuit between a classical encoding network and a classical decoding network. The encoder maps $x\in[0,1]$ 6 into features $x\in[0,1]$ 7, repeated data re-uploading encodes them through alternating $x\in[0,1]$ 8 and $x\in[0,1]$ 9 rotations before each variational layer, expectation values of Pauli- $(x,y,z)$ 0 are measured on each qubit, and a classical decoder maps the resulting vector to the scalar solution $(x,y,z)$ 1 (Klement et al., 21 Jan 2026).

The fourth is the classical backbone with quantum hidden layer. For 2D time-dependent Maxwell equations in TE $(x,y,z)$ 2 form, a classical PINN backbone performs periodic coordinate preprocessing, learned periodic time mapping, random Fourier feature embedding, and several classical hidden layers. The second-last classical layer is then replaced by a seven-qubit PQC; Pauli- $(x,y,z)$ 3 expectation values from each qubit are fed into a final classical layer predicting $(x,y,z)$ 4 (Chen et al., 29 Jun 2025). This model is explicitly hybrid and fully differentiable in simulation.

The fifth is the trainable-embedding quantum-assisted PINN. In FNN-TE-QPINN, a classical feed-forward network generates trainable angles

$(x,y,z)$ 5

for the encoding unitary

$(x,y,z)$ 6

after which a hardware-efficient variational circuit acts and the final scalar output is read out as

$(x,y,z)$ 7

In QNN-TE-QPINN, the embedding stage is itself quantum (Tran et al., 16 Feb 2026). An analogous extended framework for reaction-diffusion systems supports both classical and quantum embeddings while keeping the variational ansatz and loss fixed (Tran et al., 10 Feb 2026).

The sixth is the additive–multiplicative coupling architecture. A purely theoretical proposal, QPINN-MAC, couples a classical MLP output $(x,y,z)$ 8 with a QNode expectation value $(x,y,z)$ 9 through

$(u,v,w,p)$ 0

This retains explicit classical and quantum branches while making the quantum term both additive and multiplicative (Lantigua et al., 10 Nov 2025).

3. Physics-informed objectives and constraint enforcement

Across HQPINNs, the physics-informed component is structurally orthodox: PDE or ODE residuals are computed at collocation points, and the total objective combines residual, boundary, and initial-condition terms when needed.

For the minimal ODE HQPINN, the residual is

$(u,v,w,p)$ 1

and the loss is the mean-squared residual

$(u,v,w,p)$ 2

There is no supervised data term and no initial-condition penalty because the architecture satisfies $(u,v,w,p)$ 3 exactly (Soni, 17 Sep 2025).

For steady incompressible CFD in the Y-shaped mixer, the governing equations are

$(u,v,w,p)$ 4

which are enforced through separate residuals for the three momentum equations and continuity, summed into

$(u,v,w,p)$ 5

plus wall, inlet, and outlet boundary losses (Sedykh et al., 2023).

For parabolic PDEs, the generic residual formulation is

$(u,v,w,p)$ 6

with corresponding boundary and initial residuals

$(u,v,w,p)$ 7

The trainable-embedding QPINNs optimize

$(u,v,w,p)$ 8

where the surrogate itself is a quantum expectation value (Tran et al., 16 Feb 2026).

For nonlinear PDE benchmarks in Burgers, Allen–Cahn, and KdV, the HQPINN uses the unweighted sum

$(u,v,w,p)$ 9

with derivatives such as $(x,t)$ 0, $(x,t)$ 1, $(x,t)$ 2, and $(x,t)$ 3 obtained by autodiff through the full hybrid model (Zabihi et al., 3 Jun 2026).

In Maxwell QPINNs, the physics loss combines residual, initial, symmetry, and energy-conservation terms: $(x,t)$ 4 The added energy term is derived from Poynting’s theorem and is used to stabilize training in the source-free vacuum case (Chen et al., 29 Jun 2025).

Not all HQPINNs are PDE solvers in the narrow numerical-analysis sense. Seismic inversion HQ-PINNs use a physics-based decoder rather than a differential operator residual: a hybrid quantum encoder predicts elastic parameters, a geophysical forward model computes reflectivity via Aki–Richards or normal-incidence relations, and the network is trained by seismic reconstruction misfit plus low-frequency regularization (Vashisth et al., 6 Mar 2025). Hydrological HQC-PINNs similarly combine classification loss with differentiable PDE-constrained hydrological regularizers rather than solving a PDE field directly. Their objective is

$(x,t)$ 5

where $(x,t)$ 6 is a Saint-Venant continuity residual and $(x,t)$ 7 enforces Manning consistency (Hewage et al., 10 Apr 2026).

4. Quantum representations, embeddings, and differentiability

A central distinction across HQPINNs is whether the quantum component is fixed, trainable, or merely quantum-inspired.

The simplest case is fixed quantum feature extraction. In the ODE proof of concept, the circuit is not variational, has no trainable quantum parameters, and gradients do not pass through the quantum branch. Quantum outputs are generated first, standardized as a feature matrix $(x,t)$ 8, and then treated as fixed classical inputs (Soni, 17 Sep 2025). This is hybrid in the sense of “quantum feature map plus classical PINN,” not in the sense of joint quantum-classical optimization.

Trainable variational circuits appear in most later HQPINNs. The Maxwell architecture uses a seven-qubit PQC with 4 variational layers, angle-embedded latent activations, and several candidate ansätze, including Basic Entangling Layers, Strongly Entangling Layers, Cross-Mesh variants, and a No Entanglement control (Chen et al., 29 Jun 2025). The nonlinear-PDE HQPINN uses a hardware-efficient ansatz in which a bounded classical latent vector

$(x,t)$ 9

is angle-encoded through

$u(x,t)$ 0

followed by $u(x,t)$ 1 layers of trainable single-qubit $u(x,t)$ 2 rotations and a ring of CNOT gates (Zabihi et al., 3 Jun 2026).

Trainable embeddings have become a major research direction. FNN-TE-QPINN generates encoding angles with a small classical network, whereas QNN-TE-QPINN uses an auxiliary quantum model to generate the embedding itself (Tran et al., 16 Feb 2026). For multi-species reaction–diffusion systems, the trainable embedding is

$u(x,t)$ 3

and the encoding operator is

$u(x,t)$ 4

The resulting state is then processed by a shared variational circuit, and multiple PDE fields are read out through species-specific observables

$u(x,t)$ 5

This suggests that in multi-field HQPINNs the measurement operator itself can encode task structure, not just the circuit architecture (Tran et al., 10 Feb 2026).

Continuous-variable variants use qumodes rather than qubits. In these models, inputs are encoded by displacement gates, Gaussian gates perform affine-like transformations, and non-Gaussian Kerr gates provide the analogue of activation functions. The output is typically a quadrature expectation value such as

$u(x,t)$ 6

or multi-output variants for $u(x,t)$ 7 and its derivatives (Markidis, 2022, Panichi et al., 15 Mar 2025, Dehaghani et al., 2024).

Differentiability strategy is a major practical separator. Some architectures rely on classical autodiff through a simulator, including the Poisson and heat QPINNs implemented with Strawberry Fields and TensorFlow (Markidis, 2022, Panichi et al., 15 Mar 2025), the CFD HQPINN using adjoint differentiation on a classical simulator (Sedykh et al., 2023), and the Maxwell QPINN using a custom GPU-accelerated differentiable simulator (Chen et al., 29 Jun 2025). Other works explicitly use parameter-shift for quantum parameters while using classical backpropagation or hybrid autodiff for embedding and post-processing, as in the trainable-embedding parabolic and reaction–diffusion frameworks (Tran et al., 16 Feb 2026, Tran et al., 10 Feb 2026) and the hydrological HQC-PINN (Hewage et al., 10 Apr 2026).

A nearby but distinct line uses quantum-inspired rather than genuinely quantum computation. Battery SOH prognosis with a “QPINN” uses a fixed quantum feature mapping defined through an $u(x,t)$ 8-qubit kernel

$u(x,t)$ 9

approximated by a Nyström embedding

$\hat y(x)=1+x\,g_\theta([x,q(x)]),$ 0

This feature map is detached and consumed by a classical physics-informed model; there is no explicit variational circuit, no quantum hardware backend, and no trainable quantum parameters (Hossain et al., 11 Apr 2026). By strict usage, this is adjacent to HQPINNs rather than a canonical instance.

5. Empirical regimes, reported performance, and interpretation

Published HQPINN results do not support a single blanket claim of superiority. Instead, they suggest regime dependence.

For very simple ODEs, feasibility is established but advantage is not. The two-qubit ODE model “successfully reproduced the analytical solution $\hat y(x)=1+x\,g_\theta([x,q(x)]),$ 1” with only minor deviations at larger $\hat y(x)=1+x\,g_\theta([x,q(x)]),$ 2, but no classical baseline, no numerical error table, and no scaling study are reported (Soni, 17 Sep 2025). This is best interpreted as proof of concept.

For CFD in complex shapes, the 3D Y-mixer HQPINN reports a 21% improvement over a matched classical mini-batch model in terms of loss value or “21% higher accuracy,” depending on phrasing, while preserving the standard Navier–Stokes PINN formulation (Sedykh et al., 2023). However, the hybrid model still exhibits asymmetry between physically symmetric pipes and is trained only on a simulator, so the result indicates potential rather than CFD-grade validation.

For trainable-embedding parabolic PDEs, the evidence is more granular. On the 1D heat equation, after 150 epochs the reported losses are

$\hat y(x)=1+x\,g_\theta([x,q(x)]),$ 3

$\hat y(x)=1+x\,g_\theta([x,q(x)]),$ 4

$\hat y(x)=1+x\,g_\theta([x,q(x)]),$ 5

The best maximum absolute error is

$\hat y(x)=1+x\,g_\theta([x,q(x)]),$ 6

and the 1D inference metrics favor the classical-embedding hybrid over both the classical PINN and the fully quantum embedding (Tran et al., 16 Feb 2026). In 2D heat, FNN-TE-QPINN again dominates training loss and error, while QNN-TE-QPINN is far more parameter-efficient but notably less accurate: $\hat y(x)=1+x\,g_\theta([x,q(x)]),$ 7 A plausible implication is that embedding design governs trainability more strongly than the mere presence of a quantum variational block.

The multi-species reaction–diffusion study reaches a similar conclusion. FNN-TE-QPINN generally achieves the lowest final loss, whereas QNN-TE-QPINN often converges faster and more smoothly but saturates earlier at a higher error floor (Tran et al., 10 Feb 2026). This reinforces the view that fully quantum embeddings can be compact and stable, but classical trainable embeddings remain stronger in final approximation quality under current simulation constraints.

For nonlinear PDEs, the case for HQPINNs is more substantial. In Burgers’ equation, relative $\hat y(x)=1+x\,g_\theta([x,q(x)]),$ 8 error decreases from

$\hat y(x)=1+x\,g_\theta([x,q(x)]),$ 9

$g_\theta$ 0

about a fourfold reduction. In Allen–Cahn, the reduction is from

$g_\theta$ 1

$g_\theta$ 2

about fivefold. In KdV, the gain is more modest, from

$g_\theta$ 3

$g_\theta$ 4

These results are accompanied by smoother training dynamics and reduced loss oscillations, with the most consistent benefit appearing when the PDE exhibits stiffness, sharp gradients, localized interfaces, or multiscale structure (Zabihi et al., 3 Jun 2026). The same study reports that output-stage placement of the PQC is consistently best, and that optimal qubit count and depth are problem dependent rather than monotone with model size.

Maxwell QPINNs show that hybridization can improve both accuracy and parameter efficiency on a coupled unsteady PDE. The best settings yield up to 19% lower $g_\theta$ 5 error than the classical PINN while using about 19% fewer trainable parameters, with the quantum model inserted as a hidden layer rather than replacing the full network (Chen et al., 29 Jun 2025). However, this advantage depends strongly on input-angle scaling, circuit ansatz, and the inclusion of a global energy term in the vacuum case.

Evidence from scientific inverse problems is mixed but informative. The seismic HQ-PINN estimates impedances accurately on synthetic and field data, with ansatz choice $g_\theta$ 6, $g_\theta$ 7, or $g_\theta$ 8 making little difference and even a “No Ansatz” configuration remaining effective (Vashisth et al., 6 Mar 2025). By contrast, the high-speed-flow HQPINN shows clear benefits only for harmonic or smooth regimes. On a smooth 1D Euler traveling wave, the quantum and hybrid architectures are strongly parameter-efficient and accurate; on a moving contact discontinuity and on 2D transonic aerofoil flow, classical PINNs remain more reliable, and under-parameterized quantum branches degrade trainability (Leong et al., 4 Mar 2025). This suggests that the Fourier-like inductive bias of small PQCs is advantageous for harmonic structure but poorly matched to discontinuities.

A separate hydrological HQC-PINN reports convergence in roughly 3x fewer epochs and about 44% fewer trainable parameters than a classical PINN while maintaining competitive classification accuracy. The best $g_\theta$ 9-qubit, $q(x)$ 0-layer model reaches $q(x)$ 1 accuracy with $q(x)$ 2 parameters versus $q(x)$ 3 and $q(x)$ 4 parameters for the classical PINN, and a transfer-learning variant reaches $q(x)$ 5 (Hewage et al., 10 Apr 2026). Because this model is a physics-constrained classifier rather than a direct PDE field solver, it broadens the operational meaning of HQPINN beyond forward PDE approximation.

6. Theory, trainability, and open issues

Theoretical work has begun to formalize hybridization, but the literature remains uneven between rigorous theory and empirical validation.

QPINN-MAC provides one of the few explicit universal-approximation and trainability theorems for a hybrid ansatz. Its defining componentwise form,

$q(x)$ 6

is proved dense in $q(x)$ 7 under the stated assumptions, and the paper derives an algebraic gradient bound

$q(x)$ 8

as a mitigation against barren-plateau-style gradient collapse (Lantigua et al., 10 Nov 2025). This establishes a theoretical pathway for coupling classical and quantum branches so that the classical factor modulates the quantum gradient path. Yet the paper is theory-only: it contains no experiments, no optimizer comparisons, and no PDE benchmarks.

A complementary explanatory study argues that the practical advantage of quantum-enhanced PINNs lies primarily in faster convergence rather than markedly better asymptotic accuracy. On a family of 1D diffusion–convection PDEs, serial classical–quantum–classical qPINNs often require one to two orders of magnitude fewer epochs than parameter-matched classical PINNs to reach a given MSE, particularly when the attainable error floor is low and collocation data are sparse (Klement et al., 21 Jan 2026). This suggests that optimization efficiency rather than raw expressivity may be the principal near-term value proposition of HQPINNs.

Trainability pathologies remain a recurrent theme. Continuous-variable QPINNs for Poisson report that plain SGD outperforms Adam, Nadam, RMSprop, Adadelta, SPSA, and SGD+L-BFGS-B, reversing common classical PINN heuristics and highlighting that optimizer behavior does not transfer cleanly from classical to quantum settings (Markidis, 2022). The Maxwell work identifies a distinct “black hole” loss-landscape phenomenon, in which the quantum model initially learns useful dynamics but later collapses to an almost-zero solution for $q(x)$ 9; adding the energy conservation term removes this failure mode in the vacuum case (Chen et al., 29 Jun 2025). A plausible implication is that global physical invariants may serve both as regularizers and as trainability stabilizers in HQPINNs.

Several misconceptions are addressed by the current literature. First, hybridization does not imply a trainable quantum model; fixed quantum feature maps already satisfy a weak notion of HQPINN (Soni, 17 Sep 2025). Second, more quantum content is not uniformly better. In trainable-embedding studies, the most quantum-centric architectures are more compact but usually less accurate than hybrids with classical embeddings (Tran et al., 16 Feb 2026, Tran et al., 10 Feb 2026). Third, quantum circuits are not automatically beneficial on hard PDEs. High-speed-flow experiments show that Fourier-like quantum inductive bias can be actively harmful near discontinuities or shocks (Leong et al., 4 Mar 2025). Fourth, most reported gains are simulator-only and do not establish hardware-era advantage. Nearly all cited studies use noiseless or classical-simulator settings; shot noise, hardware noise, calibration drift, and finite-shot gradient estimation remain largely unexplored (Sedykh et al., 2023, Zabihi et al., 3 Jun 2026, Chen et al., 29 Jun 2025, Hewage et al., 10 Apr 2026).

The field also has persistent reproducibility limitations. Common omissions include explicit qubit counts, gate-level circuit definitions, optimizer hyperparameters, batch sizes, collocation-point counts, loss-weight schedules, software versions, hardware environments, and full baseline protocols. This is particularly pronounced in early proof-of-concept papers (Soni, 17 Sep 2025, Sedykh et al., 2023, Dehaghani et al., 2024). As a result, many current HQPINN results should be interpreted as architectural evidence rather than mature numerical benchmarks.

Taken together, the literature indicates that HQPINNs are most credible when the quantum component is tightly co-designed with the classical backbone, the encoding interface, and the physics-informed objective. The strongest empirical support currently favors hybrid latent processing and trainable classical embeddings for smooth, stiff, or multiscale PDEs, while discontinuous and shock-dominated regimes remain challenging and often still favor classical PINNs (Zabihi et al., 3 Jun 2026, Leong et al., 4 Mar 2025). Future work will likely determine whether these gains persist under realistic hardware constraints, whether quantum modules can outperform simply enlarging the classical backbone, and whether physics-based global constraints can systematically regularize hybrid optimization landscapes (Chen et al., 29 Jun 2025, Lantigua et al., 10 Nov 2025).