Quantum Algorithm for Estimating Intrinsic Geometry

TL;DR

This paper introduces a quantum algorithm that efficiently estimates local intrinsic dimension and scalar curvature of high-dimensional noisy datasets, enabling advanced geometric data analysis with exponential speedup over classical methods.

Contribution

The work presents the first quantum algorithm for local geometric property estimation, extending quantum manifold learning techniques to include curvature and dimension inference.

Findings

Achieves exponential speedup over classical algorithms for local geometry estimation.

Extends quantum diffusion maps with exponential improvements.

Enables scalable quantum geometric inference for noisy high-dimensional data.

Abstract

High-dimensional datasets typically cluster around lower-dimensional manifolds but are also often marred by severe noise, obscuring the intrinsic geometry essential for downstream learning tasks. We present a quantum algorithm for estimating the intrinsic geometry of a point cloud -- specifically its local intrinsic dimension and local scalar curvature. These quantities are crucial for dimensionality reduction, feature extraction, and anomaly detection -- tasks that are central to a wide range of data-driven and data-assisted applications. In this work, we propose a quantum algorithm which takes a dataset with pairwise geometric distance, output the estimation of local dimension and curvature at a given point. We demonstrate that this quantum algorithm achieves an exponential speedup over its classical counterpart, and, as a corollary, further extend our main technique to diffusion maps, yielding exponential improvements even over existing quantum algorithms. Our work marks another step toward efficient quantum applications in geometrical data analysis, moving beyond topological summaries toward precise geometric inference and opening a novel, scalable path to quantum-enhanced manifold learning.