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.
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.
We introduce an algorithm for computing geodesics on sampled manifolds that relies on simulation of quantum dynamics on a graph embedding of the sampled data. Our approach exploits classic results in semiclassical analysis and the quantum-classical correspondence, and forms a basis for techniques to learn the manifold from which a dataset is sampled, and subsequently for nonlinear dimensionality reduction of high-dimensional datasets. We illustrate the new algorithm with data sampled from model manifolds and also by a clustering demonstration based on COVID-19 mobility data. Finally, our method reveals interesting connections between the discretization provided by data sampling and quantization.
Recent advancements in the discipline of quantum algorithms have displayed the importance of the geometry of quantum operators. Given this thrust, this paper develops a rigorous geometric framework to analyze how the Riemannian structure of data, under the manifold hypothesis, influences the subspace of quantum gates induced by quantum feature maps. While numerous encoding schemes have been proposed in quantum machine learning, little attention has been given to how data geometry is deformed when mapped into the Lie group of special unitary operators. Addressing this gap, we assume a point cloud forms a smooth Riemannian manifold and formally construct the induced Riemannian geometry of a broad class of Hamiltonian quantum feature maps, which encompasses the majority of derived schemes. Starting from first principles, we derive analytic and, consequently, computational formulas for fundamental geometric measurements, including curvature, volume forms, and harmonic maps, providing tools for systematic deformation analysis. Notably, the derivations of the formulas elucidates how changes along paths in the data manifold interplay to changes in the associated subspace of special unitary operators, thereby indicating a direct geometric effect of data on quantum circuits. This framework establishes the mathematical validity required for principled analysis beyond heuristic ansatz and enables future research into geometry-aware quantum algorithm design.
Quantum topology, a rapidly evolving field, offers a fundamentally new approach to geometric modeling and data analysis. This paper explores the application of quantum topology to high-dimensional data, aiming to uncover hidden patterns and relationships within complex datasets. We propose a novel geometric modeling technique predicated on the inherent topological properties of quantum systems, leveraging these properties to enhance data representation and facilitate insightful analysis. The core claim of this research centers on the development of a robust and adaptable method for analyzing high-dimensional data, specifically designed to exploit the potential of quantum geometry. The paper details the underlying mathematical framework, provides a preliminary implementation of the core mechanism, and presents initial results demonstrating the efficacy of this approach in a simplified, illustrative example. The investigation highlights the potential for quantum topology to offer a paradigm shift in data analysis, moving beyond conventional methods and embracing the inherent complexities of high-dimensional data.
Abstract We present a rigorous mathematical analysis of the Quantum-Inspired Local Tensor Regression (QILTR) framework for modeling non-stationary, multi-linear systems. QILTR's central hypothesis is that weighting local regression problems by the Bures distance—a metric derived from the Quantum Fisher Information (QFI)—improves optimization properties. We formalize this by recasting QILTR as a Riemannian optimization problem on the manifold of quantum states, establishing that the QILTR update approximates a natural gradient descent step measured by the QFI metric. Under local geodesic convexity, we derive a formal convergence rate and demonstrate that QFI-based weighting produces a better-conditioned local Hessian matrix compared to standard Euclidean kernels. We analyze low-rank tensor estimation via weighted Alternating Least Squares (ALS), establishing the regularizing properties of quantum-geometric weights in this non-convex setting, and prove computational tractability with polynomial-time complexity per centroid under dual constraints of low-dimensional quantum encoding and low-rank coefficient tensors. Comprehensive empirical validation on synthetic datasets (150 trials, five scenarios) and real-world quantum chemistry data (QM9, ten trials) confirms that the bounded hyperbolic tangent scaling function prevents outlier amplification, yielding a 4.9× improvement. We transparently report that theoretical convergence rate advantages are not empirically observable, as both QILTR and Euclidean-LTR converge rapidly. Practical benefits manifest as improved solution quality (+1.22% on QM9). By grounding QILTR in information geometry with empirical rigor, we position it as a geometrically optimized and computationally viable scientific machine learning tool with validated design principles and realistic expectations.
My research has three main foci, one which is of interest to the fields of quantum information and quantum optics is to characterize the effects of coupling between quantum systems. Advances in technology made since the conception of quantum mechanics spurred the development of quantum technologies like quantum optics and quantum computing. These applications require a good understanding of the quantum systems that compose them, and the clever ways in which the information in these systems is stored and manipulated. By understanding the ways which quantum systems interact with their environment, their usefulness can be further utilized. The second is to develop methods of calculation which simplify the calculational and computational side of quantum dynamics. Understanding how quantum systems evolve in time is very important, but traditional methods of solving for the dynamics of quantum systems struggle to deal with time dependent parameters. Using results from the theory of Lie algebras, I will show how quantum dynamics problems can be made less computationally intensive by avoiding the partial differential equations typically associated with such problems and reducing the problem to a system of coupled ordinary differential equations. The third focus is to develop methods of visualizing properties of quantum systems. I think this is important as an educational tool and as a way to develop intuition about how quantum systems evolve considering how few opportunities there are to visualize quantum effects on a macroscopic scale. This problem is particularly challenging because quantum mechanics is non-deterministic, meaning that the only nontrivial definitive statements that can be made regarding quantum systems deal with probabilities. This coupled with the fact that many of the systems of interest depend on upwards of six variables and also evolve in time make creating useful visual representations challenging. Despite the challenges, I feel that useful insights can be gained from the visualizations, and will illustrate a few examples.
Fractal dimensionality reduction is a crucial task in many scientific domains, particularly in image processing, data analysis, and computational modeling. Traditional methods often struggle with high-dimensional data, leading to information loss and computational bottlenecks. This paper introduces a novel algorithm, "Quantum Entanglement-Enhanced Dimensionality Reduction," that leverages the principles of quantum entanglement to achieve a significant reduction in dimensionality while maintaining a high degree of fidelity. We propose a technique where entangled qubits are strategically utilized to identify and isolate key features within the data, effectively filtering out irrelevant information and preserving crucial structural elements. The core mechanism centers around creating a 'quantum feature map' – a representation where the data's structure is encoded through the entanglement of qubits. This allows for a more targeted and efficient dimensionality reduction process compared to existing approaches. We demonstrate the effectiveness of this algorithm through a series of rigorous experiments on a dataset of complex geometric shapes, showcasing its superior performance and robustness. The study highlights the potential of entanglement as a transformative resource for dimensionality reduction, offering a fundamentally new perspective on this computationally intensive problem.