Quantum Algorithm for Computing Distances Between Subspaces

TL;DR

This paper introduces a quantum algorithm that efficiently estimates distances between subspaces, such as Grassmann and ellipsoid distances, with exponential speedup under certain conditions, advancing quantum data analysis techniques.

Contribution

The paper presents the first quantum algorithm for computing subspace distances, achieving exponential speedup compared to classical methods under specific assumptions.

Findings

Quantum algorithm estimates Grassmann and ellipsoid distances.

Exponential speedup over classical algorithms under certain conditions.

Extensions to other distance measures are discussed.

Abstract

Geometry and topology have generated impacts far beyond their pure mathematical primitive, providing a solid foundation for many applicable tools. Typically, real-world data are represented as vectors, forming a linear subspace for a given data collection. Computing distances between different subspaces is generally a computationally challenging problem with both theoretical and applicable consequences, as, for example, the results can be used to classify data from different categories. Fueled by the fast-growing development of quantum algorithms, we consider such problems in the quantum context and provide a quantum algorithm for estimating two kinds of distance: Grassmann distance and ellipsoid distance. Under appropriate assumptions and conditions, the speedup of our quantum algorithm is exponential with respect to both the dimension of the given data and the number of data points. Some extensions regarding estimating different kinds of distance are then discussed as a corollary of our main quantum algorithmic method.