Certain families of quantum mechanical models can be described and solved efficiently on a classical computer, including qubit or qudit Clifford circuits and stabilizer codes, free-boson or free-fermion models, and certain rotor and GKP codes. We show that all of these families can be described as instances of the same algebraic structure, namely quadratic functions over abelian groups, or more generally over (super) Hopf algebras. Different kinds of degrees of freedom correspond to different "elementary" abelian groups or Hopf algebras:
Z2 for qubits,
Zd for qudits,
R for continuous variables, both
Z and
R/Z for rotors, and a super Hopf algebra
F for fermionic modes. Objects such as states, operators, superoperators, or projection-operator valued measures, etc, are tensors. For the solvable models above, these tensors are quadratic tensors based on quadratic functions. Quadratic tensors with
n degrees of freedom are fully specified by only
O(n2) coefficients. Tensor networks of quadratic tensors can be contracted efficiently on the level of these coefficients, using an operation reminiscent of the Schur complement. Our formalism naturally includes models with mixed degrees of freedom, such as qudits of different dimensions. We also use quadratic functions to define generalized stabilizer codes and Clifford gates for arbitrary abelian groups. Finally, we give a generalization from quadratic (or 2nd order) to
ith order tensors, which are specified by
O(ni) coefficients but cannot be contracted efficiently in general.
This paper presents a rigorous mathematical and ontological synthesis of solvable quantum systems and emergent fractal geometries. By leveraging the quadratic tensor framework, we demonstrate that disparate families of quantum models—including Clifford circuits, stabilizer codes, Gaussian states, and free-fermion systems—can be unified as quadratic functions over abelian groups or super-commutative Hopf algebras. This algebraic unification permits classical simulation with
O(n2) complexity via a generalized Schur complement contraction mechanism, dissolving the boundaries between discrete-variable qudits and continuous-variable bosonic modes. However, physical reality frequently transcends these smooth, integer-dimensional algebraic bases. We integrate a fractal-recursive ontology, identifying that quantum trajectories in path integrals and energy spectra in quantum materials (e.g., Hofstadter's butterfly) exhibit non-integer Hausdorff dimensions and self-similar architectures that challenge traditional linear causality. We further explore the transition from quadratic to
i-th order tensors, identifying the Clifford hierarchy's correspondence to higher-order polynomial maps and the resulting computational boundaries of the Gottesman-Knill theorem. Also available on Academia.edu: https://www.academia.edu/attachments/131609415/download_file?s=portfolio
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
This fifth volume completes the core mathematical arc of the series by lifting the classical information-geometric structure on the centre Z(M) (Volume II) to a full quantum Riemannian geometry on the state space of the emergent type-III algebra M. We construct a quantum Fisher information metric on the manifold of normal states of M that is compatible with the direct-integral decomposition and reduces to the classical Fisher metric on the centre. The resulting Riemannian structure is shown to be invariant under the modular flow and to encode the fractal dimension of the avalanche tree (Volume IV) as a curvature invariant. Connections are drawn to recent developments in quantum information geometry on non-commutative algebras and to the programme of fractal quantum states. The unified Riemannian picture furnishes a geometric characterisation of the algebraic Heisenberg cut as the locus where the quantum metric degenerates to its classical projection. All constructions remain anchored in the avalanche-photodiode model while providing a coherent geometric language that unifies the algebraic, modular, fractal and information-geometric threads developed across the series. K 1 S Also available on Academia.edu: https://www.academia.edu/attachments/133072858/download_file?s=portfolio
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
This volume systematically reinterprets the security foundations of lattice-based post-quantum cryptography (PQC) for the theoretical post-post-quantum era through the mathematical lens of fractal geometry and fractal algebra. The analysis asserts that cryptographic hardness boundaries are not smooth Euclidean manifolds but complex fractal objects exhibiting scale invariance. Specifically, the research demonstrates four key fractal phenomena: 1) The distribution of viable short vectors in Learning With Errors (LWE) lattices possesses a non-integer box-counting dimension (typically KSCCN Mamun, SM (2026) ……………………………………….1 Theoretical and Physical Boundaries of Lattice-Based Post-Quantum Cryptography: An Anatomical Critique of Fractal Geometry Interpretations of Post-Post-Quantum Cryptography (Vol-VIII) K S C 1.8–2.3), yielding a tighter, scale-invariant bound on average-case hardness against quantum oracles. 2) Heavy-tailed Lévy-stable noise generates scale-free random walks (Lévy flights) with infinite Hausdorff measure, geometrically overwhelming the finite Euclidean capacity of Gottesman-Kitaev-Preskill (GKP) quantum error-correcting codes at every resolution. 3) The physical limitations of quantum attacks are bounded by thermodynamic strange attractors governing Quantum Random Access Memory (QRAM) erasure costs. 4) Noisy Intermediate-Scale Quantum (NISQ) variational algorithms fail due to chaotic optimization paths across non-integer dimensional barren plateaus. Also available on Academia.edu: https://www.academia.edu/attachments/132780584/download_file?s=portfolio