Quantum Information Geometry and Machine-Learning Renormalisation: Reinterpreting the Continuous Unitary Transformations of the Similarity Renormalization Group as Geodesics on Manifolds of Quantum States and as Layers of Disentangling Deep Networks Volume: IV — Independent Study
N C K SC K SC Recent advances at the intersection of quantum information geometry and machine-learning approaches to the renormalisation group suggest a fertile reinterpretation of the Similarity Renormalization Group (SRG). The continuous unitary transformations that drive a Hamiltonian toward band-diagonal form may be viewed as geodesics on a manifold of quantum states (or of density operators) equipped with a monotone metric, or equivalently as successive layers of a deep network that progressively disentangle degrees of freedom. This independent study develops that correspondence. It reviews the information-geometric structures relevant to quantum field theories and many-body Hamiltonians, examines neural-network renormalisation-group (NN-RG) architectures, and maps the SRG flow—originally clarified in the pedagogical setting of Szpigel & Perry (2000)—onto both geometric and learning-theoretic languages. The resulting perspective yields analytic insights into the optimality of particular generators and practical algorithms for approximate Hamiltonian diagonalisation that combine SRG ideas with modern differentiable programming. Error-scaling diagnostics originating in the two-dimensional laboratory are re-expressed as statements about geodesic deviation and about generalisation error in the associated learning problem. Also available on Academia.edu: https://www.academia.edu/attachments/133749950/download_file?s=portfolio
N C K SC K SC Recent advances at the intersection of quantum information geometry and machine-learning approaches to the renormalisation group suggest a fertile reinterpretation of the Similarity Renormalization Group (SRG). The continuous unitary transformations that drive a Hamiltonian toward band-diagonal form may be viewed as geodesics on a manifold of quantum states (or of density operators) equipped with a monotone metric, or equivalently as successive layers of a deep network that progressively disentangle degrees of freedom. This independent study develops that correspondence. It reviews the information-geometric structures relevant to quantum field theories and many-body Hamiltonians, examines neural-network renormalisation-group (NN-RG) architectures, and maps the SRG flow—originally clarified in the pedagogical setting of Szpigel & Perry (2000)—onto both geometric and learning-theoretic languages. The resulting perspective yields analytic insights into the optimality of particular generators and practical algorithms for approximate Hamiltonian diagonalisation that combine SRG ideas with modern differentiable programming. Error-scaling diagnostics originating in the two-dimensional laboratory are re-expressed as statements about geodesic deviation and about generalisation error in the associated learning problem. Also available on Academia.edu: https://www.academia.edu/attachments/133749950/download_file?s=portfolio
Hilbert spaces, introduced in the early twentieth century as infinite-dimensional generalisations of Euclidean space, provide the rigorous mathematical framework for quantum mechanics and numerous applied disciplines. This paper offers a comprehensive examination of their historical development, axiomatic structure, and multifaceted applications, with particular emphasis on their foundational role in quantum mechanics. We then connect these structures to quantum information geometry, where the (projective) Hilbert space is endowed with natural Riemannian metrics—most notably the Fubini–Study metric for pure states and the Bures metric for mixed states—that quantify statistical distinguishability, parameter estimation precision, and geometric phases. By exploring classical information geometry as a precursor and highlighting quantum extensions, we illustrate how Hilbert-space geometry reveals deep insights into quantum phase transitions, metrology, and quantum technologies. Nuances such as separability, rigged Hilbert spaces for continuous spectra, and infinite-dimensional subtleties are addressed, alongside implications for machine learning via reproducing kernel Hilbert spaces (RKHS) and beyond. The analysis underscores the enduring interplay between pure mathematics and quantum physics. Mamun, SM (2026) …………………………………..1 Hilbert Spaces: Foundations, Applications, and Geometric Connections to Quantum Information Theory (Vol-I) Also available on Academia.edu: https://www.academia.edu/attachments/132277960/download_file?s=portfolio
The renormalization group (RG) is an essential technique in statistical physics and quantum field theory, which considers scale-invariant properties of physical theories and how these theories' parameters change with scaling. Deep learning is a powerful computational technique that uses multi-layered neural networks to solve a myriad of complicated problems. Previous research suggests the possibility that unsupervised deep learning may be a form of RG flow, by being a layer-by-layer coarse graining of the original data. We examined this connection on a more rigorous basis for the simple example of Kadanoff block renormalization of the 2D nearest-neighbor Ising model, with our deep learning accomplished via Restricted Boltzmann Machines (RBMs). We developed extensive renormalization techniques for the 1D and 2D Ising model to provide a baseline for comparison. For the 1D Ising model, we successfully used Adam optimization on a correlation length loss function to learn the group flow, yielding results consistent with the analytical model for infinite N. For the 2D Ising model, we successfully generated Ising model samples using the Wolff algorithm, and performed the group flow using a quasi-deterministic method, validating these results by calculating the critical exponent ν. We then examined RBM learning of the Ising model layer by layer, finding a blocking structure in the learning that is qualitatively similar to RG. Lastly, we directly compared the weights of each layer from the learning to Ising spin renormalization, but found quantitative inconsistencies for the simple case of nearest-neighbor Ising models.
While Hilbert spaces remain a cornerstone of functional analysis and quantum mechanics since their formalisation in the early 20th century, the period 2023–2026 has witnessed a surge of novel theoretical refinements, experimental validations, and cross-disciplinary integrations. This update builds directly upon the foundational treatment presented in the preceding paper, emphasising connections to quantum information geometry. Key developments include Hilbert space fragmentation in many-body systems (with experimental realisation), operator-theoretic breakthroughs (e.g., transfinite fixed points and Aluthge transforms), extensions of reproducing kernel Hilbert spaces (RKHS) to machine learning and Gaussian processes, computational advances in rigged Hilbert spaces for generalised eigenfunctions, and geometric reformulations in quantum gravity and holographic duality. We explore these from algebraic, geometric, analytic, and applied perspectives, addressing nuances such as infinite-dimensional subtleties, non-Hermitian extensions, and boundary-condition independence. Implications span quantum technologies, AI, and foundational physics, with rigorous links to the Fubini–Study/Bures metrics and quantum Fisher information. Edge cases (e.g., fragmented subspaces, evolving Hilbert spaces) and future directions are highlighted for completeness. Also available on Academia.edu: https://www.academia.edu/attachments/132280253/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 Volume II extends the group-invariance framework of Nielsen and Okamura (2026) into quantum information geometry. We provide rigorous definitions for quantum states, Umegaki relative entropy and Petz f-divergences, unitary conjugation actions, the Bures metric, and monotone metrics. We establish the quantum analogues of the classical invariance theorems and derive an explicit maximal-invariant reduction for qubit pairs under simultaneous unitary conjugation: the three scalars (r₁, r₂, cos θ) from Bloch vectors. These form the direct quantum counterpart of the classical relative location + relative scale reduction. A comprehensive QuTiP simulation verifies invariance to machine precision (~10⁻¹³) and demonstrates clean collapse of the quantum relative entropy onto these three invariants. The work unifies the symmetry-reduction principle across classical f-divergences, Fisher–Rao distance, quantum relative entropy, and Bures geometry. Also available on Academia.edu: https://www.academia.edu/attachments/133246712/download_file?s=portfolio
Quantum computing has emerged as a transformative paradigm for tackling complex matrix operations that underpin applications in machine learning and quantum information. In this paper, we synthesize recent advancements in quantum computing by integrating the quantum subroutines for matrix geometric means from Liu et al. (2025), which enable efficient solutions to nonlinear matrix equations like algebraic Riccati equations, with the Chebyshev-accelerated quantum matrix multiplication framework from Li et al. (2025), offering quadratic speedups for iterative chain operations on complex matrices. Through detailed reviews of each approach, we highlight shared primitives such as block-encoding, quantum singular value transformation, Hermitianization, and polynomial approximations, demonstrating their complementarity in constructing hybrid algorithms for nonlinear iterative problems. These hybrids achieve polylogarithmic complexities in matrix dimensions while optimizing precision and conditioning dependencies, exemplified in frameworks for Riccati iterations and kernel enhancements. Applications are explored in machine learning, including geometric mean metric learning for anomaly detection in classical and quantum data, and in quantum information, such as optimal fidelity and Rényi entropy estimation with accelerated power computations for error correction. Also available on Academia.edu: https://www.academia.edu/attachments/125700844/download_file?s=portfolio