Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Advancing Quantum Matrix Computations: Integrating Geometric Means and Chain Multiplications for Nonlinear and Iterative Challenges in Machine Learning and Quantum Information

Syed Muntasir Mamun

Published Dec 11, 2025• 0 citations
Publisher

Abstract

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

DOI
10.5281/zenodo.22994162

Related papers

Quantum Synergies in Nonlinear Optimization: A Comprehensive Review and Extension of Gate-Based Genetic Algorithms and Matrix Subroutine Integrations for Quantum-Enhanced Computational Frameworks

Syed Muntasir Mamun

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

•Jan 1, 2026•0 citations

Quantum algorithms for matrix geometric means

Nana Liu, Qisheng Wang, Mark M. Wilde +1 more

Matrix geometric means between two positive definite matrices can be defined from distinct perspectives—as solutions to certain nonlinear systems of equations, as points along geodesics in Riemannian geometry, and as solutions to certain optimisation problems. We devise quantum subroutines for the matrix geometric means, and construct solutions to the algebraic Riccati equation—an important class of nonlinear systems of equations appearing in machine learning, optimal control, estimation, and filtering. Using these subroutines, we present a new class of quantum learning algorithms, for both classical and quantum data, called quantum geometric mean metric learning, for weakly supervised learning and anomaly detection. The subroutines are also useful for estimating geometric Rényi relative entropies and the Uhlmann fidelity, in particular achieving optimal dependence on precision for the Uhlmann and Matsumoto fidelities. Finally, we provide a BQP-complete problem based on matrix geometric means that can be solved by our subroutines.

arXiv:2405.00673•npj Quantum Information•Jun 13, 2025•9 citations

Faster quantum subroutine for matrix chain multiplication via Chebyshev approximation

Xinying Li, Pei-Lin Zheng, Chengkang Pan +3 more

Matrix operations are crucial to various computational tasks in various fields, and quantum computing offers a promising avenue to accelerate these operations. We present a quantum matrix multiplication (QMM) algorithm that employs amplitude encoding and combines quantum walks with Chebyshev polynomial approximation to achieve quadratic acceleration for matrix chain multiplication where the same matrix is applied K times while maintaining logarithmic complexity in matrix dimension and precision. The algorithm can be applied to any complex matrix. Furthermore, we discuss integrating our QMM algorithm as a subroutine in other matrix operations and propose strategies to optimize QMM for matrices with large condition numbers with numerical simulation.

Scientific Reports•Aug 5, 2025•3 citations

Geometric Matrix Dynamics & Quantum Computing Integration

Jincheng Zhang

This paper explores the integration of geometric matrix dynamics with quantum computing to achieve high-precision and high-efficiency geometric computations. Traditional methods often struggle to meet stringent computational requirements. This study proposes a novel algorithmic framework that leverages quantum computing to accelerate geometric matrix calculations and dynamically adjusts the matrix structure, resulting in enhanced accuracy and performance. We detail the design of this system, emphasizing the interplay between quantum acceleration and adaptive matrix manipulation. This work aims to push the boundaries of geometric computation, offering a potential paradigm shift for complex geometric problems.

•Sep 4, 2026•0 citations

Predicting Q-Day: A Fractal Mathematics Approach to Quantum Computing Timelines

Syed Muntasir Mamun

This paper employs fractal mathematics principles to forecast Q-Day—the threshold at which fault-tolerant quantum computers can practically compromise public-key cryptography—over an 18-month horizon from December 2025. Synthesizing insights from recent advancements in quantum ecosystems, algorithmic hybrids for nonlinear matrix operations, and interdisciplinary frameworks integrating quantum entanglement with fractal geometries in high-dimensional computing, the analysis models technological progress as self-similar iterative processes. Utilizing iterated function systems and fractional dimensions (approximately 1.618, akin to the golden ratio), the fractal model projects accelerated convergence toward utility-scale systems, predicting Q-Day by late 2027 with a 70% probability within the specified timeframe. Key drivers include the end of the Noisy Intermediate-Scale Quantum (NISQ) era, polylogarithmic speedups in quantum algorithms, and recursive self-improvement in autopoietic multi-agent systems. Implications encompass urgent transitions to post-quantum cryptography, geopolitical divergences in investment strategies, and existential risks from quantum-AI synergies, advocating for coordinated policies in commercialization, international collaboration, and ethical regulation to mitigate vulnerabilities while harnessing transformative potential in cybersecurity, healthcare, and environmental monitoring. Also available on Academia.edu: https://www.academia.edu/attachments/125804561/download_file?s=portfolio

•Dec 20, 2025•0 citations

Nonlinear Matrix Operations: Quantum Entanglement Enhanced Computation

Jincheng Zhang

This paper explores the potential of quantum entanglement to revolutionize matrix operations, specifically aiming to overcome the limitations of classical computers. We propose a novel approach to matrix computation that leverages entanglement to transfer the core computational process across quantum systems, thereby achieving enhanced efficiency and precision. The core claim centers on the design of a quantum-based matrix multiplication algorithm that surpasses current computational capabilities. This research investigates the theoretical basis, potential implementation challenges, and preliminary experimental explorations of this quantum approach. The paper discusses the core mechanism, its implications for computational power, and its potential impact on both quantum computing and machine learning.

•Sep 3, 2026•0 citations
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
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
Matrix geometric means between two positive definite matrices can be defined from distinct perspectives—as solutions to certain nonlinear systems of equations, as points along geodesics in Riemannian geometry, and as solutions to certain optimisation problems. We devise quantum subroutines for the matrix geometric means, and construct solutions to the algebraic Riccati equation—an important class of nonlinear systems of equations appearing in machine learning, optimal control, estimation, and filtering. Using these subroutines, we present a new class of quantum learning algorithms, for both classical and quantum data, called quantum geometric mean metric learning, for weakly supervised learning and anomaly detection. The subroutines are also useful for estimating geometric Rényi relative entropies and the Uhlmann fidelity, in particular achieving optimal dependence on precision for the Uhlmann and Matsumoto fidelities. Finally, we provide a BQP-complete problem based on matrix geometric means that can be solved by our subroutines.
Matrix operations are crucial to various computational tasks in various fields, and quantum computing offers a promising avenue to accelerate these operations. We present a quantum matrix multiplication (QMM) algorithm that employs amplitude encoding and combines quantum walks with Chebyshev polynomial approximation to achieve quadratic acceleration for matrix chain multiplication where the same matrix is applied K times while maintaining logarithmic complexity in matrix dimension and precision. The algorithm can be applied to any complex matrix. Furthermore, we discuss integrating our QMM algorithm as a subroutine in other matrix operations and propose strategies to optimize QMM for matrices with large condition numbers with numerical simulation.
This paper explores the integration of geometric matrix dynamics with quantum computing to achieve high-precision and high-efficiency geometric computations. Traditional methods often struggle to meet stringent computational requirements. This study proposes a novel algorithmic framework that leverages quantum computing to accelerate geometric matrix calculations and dynamically adjusts the matrix structure, resulting in enhanced accuracy and performance. We detail the design of this system, emphasizing the interplay between quantum acceleration and adaptive matrix manipulation. This work aims to push the boundaries of geometric computation, offering a potential paradigm shift for complex geometric problems.
This paper employs fractal mathematics principles to forecast Q-Day—the threshold at which fault-tolerant quantum computers can practically compromise public-key cryptography—over an 18-month horizon from December 2025. Synthesizing insights from recent advancements in quantum ecosystems, algorithmic hybrids for nonlinear matrix operations, and interdisciplinary frameworks integrating quantum entanglement with fractal geometries in high-dimensional computing, the analysis models technological progress as self-similar iterative processes. Utilizing iterated function systems and fractional dimensions (approximately 1.618, akin to the golden ratio), the fractal model projects accelerated convergence toward utility-scale systems, predicting Q-Day by late 2027 with a 70% probability within the specified timeframe. Key drivers include the end of the Noisy Intermediate-Scale Quantum (NISQ) era, polylogarithmic speedups in quantum algorithms, and recursive self-improvement in autopoietic multi-agent systems. Implications encompass urgent transitions to post-quantum cryptography, geopolitical divergences in investment strategies, and existential risks from quantum-AI synergies, advocating for coordinated policies in commercialization, international collaboration, and ethical regulation to mitigate vulnerabilities while harnessing transformative potential in cybersecurity, healthcare, and environmental monitoring. Also available on Academia.edu: https://www.academia.edu/attachments/125804561/download_file?s=portfolio
This paper explores the potential of quantum entanglement to revolutionize matrix operations, specifically aiming to overcome the limitations of classical computers. We propose a novel approach to matrix computation that leverages entanglement to transfer the core computational process across quantum systems, thereby achieving enhanced efficiency and precision. The core claim centers on the design of a quantum-based matrix multiplication algorithm that surpasses current computational capabilities. This research investigates the theoretical basis, potential implementation challenges, and preliminary experimental explorations of this quantum approach. The paper discusses the core mechanism, its implications for computational power, and its potential impact on both quantum computing and machine learning.