Hybrid quantum-classical framework for Betti number estimation with applications to topological data analysis

TL;DR

This paper introduces a hybrid quantum-classical algorithm for estimating Betti numbers in topological data analysis, potentially offering significant speedups over existing quantum approaches.

Contribution

It presents a novel hybrid approach combining classical enumeration of simplices with quantum estimation of Betti numbers, expanding the toolkit for TDA.

Findings

Potential polynomial to exponential speedups identified

Hybrid approach leverages classical and quantum strengths

Demonstrates utility of normalized Betti numbers in applications

Abstract

Topological data analysis (TDA) is a rapidly growing area that applies techniques from algebraic topology to extract robust features from large-scale data. A key task in TDA is the estimation of (normalized) Betti numbers, which capture essential topological invariants. While recent work has led to quantum algorithms for this problem, we explore an alternative direction: combining classical and quantum resources to estimate the Betti numbers of a simplicial complex more efficiently. Assuming the classical description of a simplicial complex, that is, its set of vertices and edges, we propose a hybrid quantum-classical algorithm. The classical component enumerates all simplices, and this combinatorial structure is subsequently processed by a quantum algorithm to estimate the Betti numbers. We analyze the performance of our approach and identify regimes where it potentially achieves polynomial to exponential speedups over existing quantum methods, at the trade-off of using more ancilla qubits. We further demonstrate the utility of normalized Betti numbers in concrete applications, highlighting the broader potential of hybrid quantum algorithms in topological data analysis.