Quantum algorithm for persistent Betti numbers and topological data analysis

TL;DR

This paper introduces the first quantum algorithm capable of efficiently estimating persistent Betti numbers across arbitrary dimensions, offering exponential speedup for high-dimensional topological data analysis.

Contribution

It presents a novel quantum algorithm for persistent Betti numbers, extending quantum topological data analysis to arbitrary dimensions.

Findings

Provides exponential speedup over classical algorithms

Efficiently estimates persistent Betti numbers in high dimensions

Applicable to complexes like Vietoris-Rips

Abstract

Topological data analysis (TDA) is an emergent field of data analysis. The critical step of TDA is computing the persistent Betti numbers. Existing classical algorithms for TDA are limited if we want to learn from high-dimensional topological features because the number of high-dimensional simplices grows exponentially in the size of the data. In the context of quantum computation, it has been previously shown that there exists an efficient quantum algorithm for estimating the Betti numbers even in high dimensions. However, the Betti numbers are less general than the persistent Betti numbers, and there have been no quantum algorithms that can estimate the persistent Betti numbers of arbitrary dimensions. This paper shows the first quantum algorithm that can estimate the (normalized) persistent Betti numbers of arbitrary dimensions. Our algorithm is efficient for simplicial complexes such as the Vietoris-Rips complex and demonstrates exponential speedup over the known classical algorithms.