Wai-Hong Tam, Reza Safari and Hiromichi Matsuyama (2026) introduce Geometric Quantum Physics-Informed Neural Networks (GQPINNs) as a symmetry-aware extension of quantum physics-informed neural networks (QPINNs). Their framework embeds finite-group (Klein four-group (K_4)) and compact Lie-group (SO(2) rotational) symmetries directly into the parametrized quantum circuit ansatz using twirling-based construction of equivariant generator sets. This builds on geometric quantum machine learning (GQML) to enforce solution-level invariance/equivariance by construction, rather than through loss penalties or post-hoc constraints. The paper benchmarks GQPINNs on representative linear and nonlinear PDEs (2D Poisson, 2D diffusion, 1D acoustic wave, 1D viscous Burgers), demonstrating superior mean absolute error (MAE) and parameter efficiency compared to baseline QPINNs and symmetry-adapted classical PINNs. Also available on Academia.edu: https://www.academia.edu/attachments/132740816/download_file?s=portfolio
Wai-Hong Tam, Reza Safari and Hiromichi Matsuyama (2026) introduce Geometric Quantum Physics-Informed Neural Networks (GQPINNs) as a symmetry-aware extension of quantum physics-informed neural networks (QPINNs). Their framework embeds finite-group (Klein four-group (K_4)) and compact Lie-group (SO(2) rotational) symmetries directly into the parametrized quantum circuit ansatz using twirling-based construction of equivariant generator sets. This builds on geometric quantum machine learning (GQML) to enforce solution-level invariance/equivariance by construction, rather than through loss penalties or post-hoc constraints. The paper benchmarks GQPINNs on representative linear and nonlinear PDEs (2D Poisson, 2D diffusion, 1D acoustic wave, 1D viscous Burgers), demonstrating superior mean absolute error (MAE) and parameter efficiency compared to baseline QPINNs and symmetry-adapted classical PINNs. Also available on Academia.edu: https://www.academia.edu/attachments/132740816/download_file?s=portfolio
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.
Computational fluid dynamics (CFD) has long been limited by the extreme computational cost and multiscale complexity of the Navier-Stokes equations, particularly in turbulent regimes. We propose that geometric quantum physics-informed neural networks (GQPINNs) — symmetry-embedded variational quantum circuits that enforce finite-group ((K_4)) and compact Lie-group (SO(2), (\mathbb{Z}_2)) invariances via twirling of generator sets — constitute the foundational architecture for next-generation quantum-inspired artificial intelligence in CFD. Drawing on the mechanics of data re-uploading circuits, group-equivariant trainable blocks, and physics-informed loss functions, we show how GQPINNs achieve orders-of-magnitude gains in parameter efficiency and solution accuracy on representative PDE benchmarks while naturally extending to the incompressible and compressible Navier-Stokes system. Ontologically, the framework realises an onto-dynamic structural realism in which fluid structures (vorticity fields, streamlines, conserved quantities) are relationally individuated by symmetry Mamun, SM (2026) …………………………………….1 Navier-Stokes Turbulence as a Computational Laboratory for Next-Generation Quantum-Inspired Artificial Intelligence: Mechanics, Ontology, and Epistemology of Geometric Quantum Physics-Informed Neural Networks (Vol-I) K S C groups rather than by intrinsic point-like properties. Also available on Academia.edu: https://www.academia.edu/attachments/132750817/download_file?s=portfolio
Quantum physics-informed neural networks (QPINNs) solve partial differential equations (PDEs) by training parameterized quantum circuits against physics-based residuals, yet the role of the embedding that maps coordinates into quantum states remains insufficiently understood. In this research, we introduce a unified embedding framework that formulates embedding as a functional transformation shaping the feature representation available to the variational circuit, and we propose two embeddings within it. The Linear Quantum Neural Network Trainable Embedding QPINN (LQNN-TE-QPINN) generates data-dependent features with an auxiliary quantum circuit and adds them to the input coordinates, instead of multiplying them as in our previous formulation. The adaptive-frequency QPINN (AdaFreq-QPINN) promotes the frequency-scaling factors of an arccos-Chebyshev encoding to trainable parameters. We evaluate both against direct and fixed analytical frequency encodings and alternative hybrid architectures on one- and two-dimensional Burgers equations. In one dimension, LQNN-TE-QPINN achieves a relative L2 error of 0.0916 with 720 trainable parameters, compared with 0.4501 for direct-encoding QPINN and 0.1233 for a locally adaptive classical physics-informed neural network with approximately eleven times more parameters, while AdaFreq-QPINN attains the lowest error among the analytical embeddings with only four additional parameters. In two dimensions, LQNN-TE-QPINN attains the lowest training objective, although its solution error is not clearly separated from that of a data re-uploading QPINN. Under simulated hardware noise, LQNN-TE-QPINN retains the lowest absolute derivative error while degrading most relative to its noiseless baseline. These results demonstrate that embedding design substantially influences the approximation and optimization behavior of QPINNs.
This report rigorously synthesizes the Geometric Quantum Physics-Informed Neural Network (GQPINN) framework, which leverages group-theoretic twirling to enforce geometric symmetries, with Fractal Algebra and the Pellis-Fractal formalism. The original GQPINN, while superior in efficiency and accuracy for linear partial differential equations (PDEs), struggles fundamentally in regimes dominated by strong dissipation, nonlinearity, or multiscale phenomena due to its reliance on smooth Lie-group symmetries. To overcome this, the novel Fractal Geometric Quantum Physics-Informed Neural Network (FGQPINN) is formulated. This architecture utilizes the Pellis-Fractal Twirling Operator (T̃(α)) to project quantum generators onto a nested hierarchy of scale-weighted commutants, effectively hard-wiring self-similar and non-differentiable symmetries into the variational quantum circuit (VQC) ansatz. Furthermore, the physics-informed loss function is adapted using the Pellis-Fractal Laplacian (ΔPF). This synthesis not only achieves scale-covariant generalization and provides a theoretical mechanism to mitigate barren plateaus by restricting entanglement dynamics Mamun, SM (2026) …………………………….1 Geometric Quantum Physics-Informed Neural Networks: A Synthesis of Ontic Structural Realism and Fractal Algebra (Vol-II) K S C but also operationalizes Ontic Structural Realism, confirming that structural relations are ontologically primary. Also available on Academia.edu: https://www.academia.edu/attachments/132750951/download_file?s=portfolio
Geometric quantum machine learning incorporates symmetry into quantum models, but how symmetry shapes their expressivity and guides effective model design remains insufficiently understood. We address this question through the Fourier representation of quantum Fourier models (QFMs). Symmetry organizes the frequency spectrum into orbits and sums the Fourier coefficients within each orbit into a symmetrized coefficient. When QFM trainable layers form independent exact 2-designs, the variance of each symmetrized coefficient equals the sum of the coefficient variances in its orbit. For single-layer QFMs with ε-approximate 2-design trainable layers, we bound the deviation from this identity. The resulting bound for individual Fourier coefficients can be exponentially tighter than an existing bound. The hyperoctahedral group provides an example of orbit growth that can mitigate vanishing expressivity of the symmetrized coefficients. Symmetrization of QFMs over an elementary abelian 2-group also yields pure multivariate Chebyshev polynomial basis functions. We introduce randomized encoding, which implements invariant models without ancilla qubits or the additional circuit depth for quantum twirling. We evaluate the models as quantum physics-informed neural networks (QPINNs) on two-dimensional screened Poisson and stationary viscous Hamilton-Jacobi equations. Under hard boundary constraints, QPINNs using exact symmetrization and randomized encoding achieve the lowest mean errors in the two benchmarks, respectively.
This paper provides a rigorous academic review of the seminal work by Mamun and Grok xAI (2025) on advancing quantum matrix computations through the integration of geometric mean subroutines and Chebyshev-accelerated chain multiplications for nonlinear and iterative challenges in machine learning and quantum information. Following a thorough summary of its salient contributions, we compare and contrast the approach with contemporary research in quantum genetic algorithms (QGAs), variational quantum eigensolvers (VQEs), and tensor network methods. Strengths in polylogarithmic complexity and hybrid fault-tolerance are highlighted, alongside limitations in NISQ-era decoherence. Extending the framework, we propose novel integrations with fractal-recursive ontologies for non-integer dimensional simulations, the Principle of Co-Adaptation for bivariational dynamics in adaptive quantum circuits, and Souriau's axiomatic geometric model for positive-type state representations in high-dimensional entanglement. These extensions unify quadratic tensor structures with emergent fractal geometries, enabling scalable applications in quantum gravity simulations and autopoietic cognitive systems. Implications for post-quantum cryptography and geopolitical quantum strategies are discussed, advocating for interdisciplinary policies to harness transformative potential while mitigating risks. Also available on Academia.edu: https://www.academia.edu/attachments/131744773/download_file?s=portfolio