Higher-order topological kernels via quantum computation

TL;DR

This paper introduces a quantum computing method for efficiently calculating higher-dimensional topological features of data, enabling advanced kernel methods in machine learning with potential quantum advantages.

Contribution

It proposes a novel quantum approach to defining topological kernels based on Betti curves, improving computational efficiency for complex topological features.

Findings

Prototype implementation on a noiseless simulator demonstrates feasibility.

Empirical results suggest quantum topological kernels may outperform classical methods.

The approach offers robustness and potential advantages in quantum machine learning.

Abstract

Topological data analysis (TDA) has emerged as a powerful tool for extracting meaningful insights from complex data. TDA enhances the analysis of objects by embedding them into a simplicial complex and extracting useful global properties such as the Betti numbers, i.e. the number of multidimensional holes, which can be used to define kernel methods that are easily integrated with existing machine-learning algorithms. These kernel methods have found broad applications, as they rely on powerful mathematical frameworks which provide theoretical guarantees on their performance. However, the computation of higher-dimensional Betti numbers can be prohibitively expensive on classical hardware, while quantum algorithms can approximate them in polynomial time in the instance size. In this work, we propose a quantum approach to defining topological kernels, which is based on constructing Betti curves, i.e. topological fingerprint of filtrations with increasing order. We exhibit a working prototype of our approach implemented on a noiseless simulator and show its robustness by means of some empirical results suggesting that topological approaches may offer an advantage in quantum machine learning.