KSCCN Series on Critical AI, Complexity & Digital Sovereignty Studies Teaching Note, Volume II: Hilbert Space in Quantum Computing Theory, Epistemology, and a Comparison of Leading Formalisms — A Companion to Harikaran M.'s Pedagogical Sketch
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
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
The intersection of quantum mechanics and mathematical optimization represents a fundamental reimagining of computational complexity, shifting the paradigm from deterministic Boolean logic to probabilistic unitary evolution within a complex Hilbert space. This report provides an exhaustive, critical analysis of the 2026 survey A Survey of Quantum Computing Algorithms for Mathematical Optimization: State-of-the-Art in Research & Exploration of Further Possibilities by Chakrabarti, Changdar, and Khanda. By contextualizing their findings within the rigorous frameworks of Hamiltonian mechanics and the geometry of time—specifically drawing upon the Anandan-Aharonov relation and the structured time geometry of Modgil—we elucidate the physical mechanisms that underpin quantum advantage. We rigorously examine the novel Quantum Collaborative Filtering (QCF) and Quantum Social Network (QSN) models proposed by the authors, deconstructing the specific Hamiltonians that drive their variational landscapes. Furthermore, we posit that the efficiency of these algorithms is governed by the geodesic curvature of the projective Hilbert space, where the "speed" of optimization is physically constrained by the energy Mamun, SM (2026) ………………………………………………….1 Hamiltonian Dynamics and the Geometry of Time: A Variational Framework for Quantum Mathematical Optimization uncertainty of the driving Hamiltonian. Also available on Academia.edu: https://www.academia.edu/attachments/131793645/download_file?s=portfolio
This comprehensive review dissects the profound philosophical tension at the core of quantum foundations: the ontological status of the Hilbert space formalism versus the abstract algebra of observables. Focusing on the Gelfand-Naimark-Segal (GNS) construction, the paper first establishes this theorem as the foundational mathematical bridge that uniquely derives a geometric Hilbert space representation from any given C*-algebra of observables and a specified physical state. The analysis then evaluates the interpretational crisis presented by the existence of unitarily inequivalent representations (UIRs) in infinite quantum systems, which challenges the ideal of a single, universal quantum state space. By comparing Algebraic Imperialism (Algebraic Quantum Field Theory) with highly sophisticated realist programs, the paper details the radical objectification of the Hilbert space. Specifically, it contrasts Wavefunction Realism (WFR), which posits a physically real field in a high-dimensional configuration space, with the minimalist ontology of Hilbert Space Realism (HSR), or "Mad-Dog Everettianism". Also available on Academia.edu: https://www.academia.edu/attachments/132963048/download_file?s=portfolio