Geometric data analysis through quantum dynamics.

Seth Lloyd, Paolo Zanardi A, Silvano Garnerone

Kumar, Akshat, Mohan Sarovar

Experimental sciences have come to depend heavily on our ability to organize and interpret high-dimensional datasets. Natural laws, conservation principles, and inter-dependencies among observed variables yield geometric structure, with fewer degrees of freedom, on the dataset. We introduce the frameworks of semiclassical and microlocal analysis to data analysis and develop a novel, yet natural uncertainty principle for extracting fine-scale features of this geometric structure in data, crucially dependent on data-driven approximations to quantum mechanical processes underlying geometric optics. This leads to the first tractable algorithm for approximation of wave dynamics and geodesics on data manifolds with rigorous probabilistic convergence rates under the manifold hypothesis. We demonstrate our algorithm on real-world datasets, including an analysis of population mobility information during the COVID-19 pandemic to achieve four-fold improvement in dimensionality reduction over existing state-of-the-art and reveal anomalous behavior exhibited by less than 1.2% of the entire dataset. Our work initiates the study of data-driven quantum dynamics for analyzing datasets, and we outline several future directions for research.

Jincheng Zhang

Quantum-Enhanced Geometric Modeling through Dynamic Embedding (QGEM-DE) presents a novel approach to geometric modeling by leveraging quantum algorithms to generate dynamic representations of data. This method aims to overcome limitations of traditional geometric models, particularly in efficiently exploring complex data spaces. The core mechanism involves encoding data into quantum states, enabling rapid exploration and potentially uncovering hidden patterns and relationships. This research explores the potential of this dynamic embedding technique to enhance geometric representation and analysis capabilities.

Liyabona manase Ngconde mdange ka tshiwo

Günter Mahler A, Volker A. Weberruß

Jincheng Zhang

This paper introduces a novel approach to quantum feature extraction based on dynamic geometric mapping. Traditional feature extraction methods often rely on classical computational techniques, limiting their ability to fully exploit the inherent quantum properties of data. We propose a new method that leverages quantum entanglement to represent and extract features, offering a fundamentally different perspective. This approach utilizes a dynamic geometric mapping to translate data into a quantum state, enabling a more nuanced and efficient extraction process. The core mechanism centers around simulating quantum feature extraction through the interplay of entangled geometric relationships. We demonstrate the efficacy of this method through theoretical analysis and preliminary experimental results, showcasing a significant improvement in feature extraction accuracy and speed compared to conventional approaches. The fundamental insight lies in harnessing quantum entanglement to guide the data transformation, providing a powerful tool for quantum data analysis.