From Betti Numbers to Persistence Diagrams: A Hybrid Quantum Algorithm for Topological Data Analysis

TL;DR

This paper tackles the obstacle that quantum topological data analysis typically yields only Betti numbers, lacking persistence diagram information. It proposes a quantum-classical hybrid pipeline that leverages harmonic-form features from the LGZ algorithm, multi-scale topological kernels, and a quantum LS-SVM to predict persistence diagrams, training on classical diagrams and then operating fully quantum at prediction time. Key contributions include elevating quantum topology from statistical summaries to pattern recognition, enabling practical persistence diagram information while retaining quantum speedups, and introducing a classical-precision-guided quantum efficiency paradigm. The approach is particularly suited to domains with finite, enumerable topological patterns and offers a feasible path toward real-time, large-scale quantum topological analysis as hardware matures.

Abstract

Persistence diagrams serve as a core tool in topological data analysis, playing a crucial role in pathological monitoring, drug discovery, and materials design. However, existing quantum topological algorithms, such as the LGZ algorithm, can only efficiently compute summary statistics like Betti numbers, failing to provide persistence diagram information that tracks the lifecycle of individual topological features, severely limiting their practical value. This paper proposes a novel quantum-classical hybrid algorithm that achieves, for the first time, the leap from "quantum computation of Betti numbers" to "quantum acquisition of practical persistence diagrams." The algorithm leverages the LGZ quantum algorithm as an efficient feature extractor, mining the harmonic form eigenvectors of the combinatorial Laplacian as well as Betti numbers, constructing specialized topological kernel functions to train a quantum support vector machine (QSVM), and learning the mapping from quantum topological features to persistence diagrams. The core contributions of this algorithm are: (1) elevating quantum topological computation from statistical summaries to pattern recognition, greatly expanding its application value; (2) obtaining more practical topological information in the form of persistence diagrams for real-world applications while maintaining the exponential speedup advantage of quantum computation; (3) proposing a novel hybrid paradigm of "classical precision guiding quantum efficiency." This method provides a feasible pathway for the practical implementation of quantum topological data analysis.