Quantum Geometry of Data

Abstract

We demonstrate how Quantum Cognition Machine Learning (QCML) encodes data as quantum geometry. In QCML, features of the data are represented by learned Hermitian matrices, and data points are mapped to states in Hilbert space. The quantum geometry description endows the dataset with rich geometric and topological structure---including intrinsic dimension, quantum metric, and Berry curvature---derived directly from the data. QCML captures global properties of data, while avoiding the curse of dimensionality inherent in local methods. The present work provides the first full exposition of QCML as a framework for \emph{matrix geometry}, establishing its mathematical foundation and demonstrating how operator-level tools from quantum geometry can be directly applied to data. We introduce the matrix Laplacian and its eigenmaps as a geometry-preserving alternative to graph-based embeddings, and we show how topological invariants such as Chern numbers can be extracted from data for the first time. An important practical outcome is the efficiency of the representation: high-dimensional datasets can be faithfully modeled with Hilbert spaces of size as small as $N \sim 10$, revealing global structures that are hard to capture with linear methods such as PCA. We illustrate these results on a number of synthetic and real-world examples, including the Wisconsin Breast Cancer dataset, where QCML uncovers geometric separability between benign and malignant samples and yields an interpretable decomposition of feature relevance. The emphasis of this work is on a new geometric representation of data rather than on algorithmic benchmarking, establishing a foundation for extracting global geometric and topological structure from high-dimensional datasets.