From Betti Numbers to Persistence Diagrams: A Hybrid Quantum Algorithm for Topological Data Analysis
This work addresses the problem of limited practical value in quantum topological data analysis for researchers and practitioners in fields such as medicine and materials science, representing a novel advancement rather than an incremental improvement.
The paper tackles the limitation of existing quantum topological algorithms that only compute Betti numbers by proposing a hybrid quantum-classical algorithm that, for the first time, enables quantum acquisition of persistence diagrams, achieving exponential speedup while providing practical topological information for applications like pathological monitoring and drug discovery.
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.