arXiv:2507.21135 · v3

Quantum Geometry of Data

首次提交 2025-07-23 05:27

版本更新 2026-11-12 00:53

雷达观测 2026-11-13 14:16

摘要

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 through a compact operator representation. At fixed Hilbert-space dimension, the number of operator parameters grows only linearly with the number of features. The present work provides the first full exposition of QCML as a framework for 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: the relevant geometric structure of high-dimensional datasets can be captured with Hilbert spaces of size as small as $N \sim 10$. 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.

作者备注：31 pages, 16 figures. Updated the Wisconsin Breast Cancer analysis, added a versioned repository reference, clarified the distinction between the response matrix and the quantum metric, and corrected formulas and simulation details