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arXiv:1701.02549 · Physica A Statistical Mechanics and its Applications

Geometric algebra and information geometry for quantum computational software

Carlo Cafaro

Published Dec 6, 2016• 31 citations
arXivHTMLPDF
DOI
10.1016/j.physa.2016.11.117

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Geometric and topological methods play an increasingly important role in quantum information science and quantum computation. Beyond the conventional Hilbert space formalism, a variety of mathematical frameworks, including group representations, mapping class groups, modular tensor categories, and character varieties, have been proposed to describe quantum states and quantum gates in a structurally robust manner. This review surveys the development of topological and geometric approaches to quantum information, with particular emphasis on representations of fundamental groups into SL(2,C), their associated character varieties, and the algebraic surfaces arising from trace coordinates, such as Fricke and Cayley cubic surfaces. These structures provide a geometric encoding of quantum degrees of freedom and offer alternative perspectives on topological quantum computing beyond anyon-based models. We also examine connections with integrable systems and isomonodromic deformations, where Painlevé equations and monodromy data supply a dynamical viewpoint on quantum state evolution. A critical comparison is provided with other geometric and topological approaches to quantum information, including geometric quantum mechanics, information geometry, tensor network geometry, and category-theoretic formulations. By synthesising results from topology, algebraic geometry, and mathematical physics, this review aims to clarify the conceptual landscape of topological quantum information geometry and to identify open problems and emerging directions in the field.
This paper explores Jean-Marie Souriau's axiomatic framework for quantum mechanics, rooted in representation theory and information geometry, as revived by Frédéric Barbaresco. It delineates quantum states as normalized positive-type functions on dynamical groups, satisfying subordination, Hermitian positivity, and quantization inequalities that yield Hilbert spaces, unitary representations, uncertainty relations, and mixed/Gibbs states. The model unifies classical symplectic geometry with quantum axiomatics, exemplified in linear Heisenberg-Weyl systems producing coherent states. Comparative analysis contrasts Souriau's approach with Kostant's geometric quantization, algebraic QM, and information-geometric metrics (e.g., Balian-Petz), highlighting its group-theoretic strengths and diffeological generality. Extensions propose integrations with infinite-dimensional QFT, quantum thermodynamics, entanglement geometry, and applications in signal processing, machine learning, and quantum computing, forging a cohesive symmetry-driven geometric paradigm for quantum phenomena. Mamun, SM (2026)………………………………..1 Souriau's Axiomatic Geometric Model of Quantum States: Representation Theory, Positive-Type Functions, Information Geometry, and Extensions Beyond Linear Systems Also available on Academia.edu: https://www.academia.edu/attachments/131647634/download_file?s=portfolio
We present a real geometric algebra framework designed for the direct translation of the Dirac formalism into geometric algebra representations. Unlike previous approaches based on positive-definite signatures, QCA employs a split-signature construction that enables a natural realization of quantum states and operators while simplifying computational implementation. We further present an implementation of QCA using the \textit{GAALOP} software and show how quantum gates and multi-qubit systems can be efficiently represented and generated computationally. As an application, we demonstrate the use of QCA in quantum game theory, where the real-algebraic formulation provides computational advantages for modeling entangled strategies and quantum interactions. The proposed framework establishes a practical bridge between the abstract formalism of quantum computation and efficient geometric algebra implementations.