New aspects of quantum topological data analysis: Betti number estimation, and testing and tracking of homology and cohomology classes

2025-06-02

AI Summary

Quantum computation for topological data analysis (TDA) faces a fundamental barrier—estimating Betti numbers is NP-hard, precluding generic quantum advantage. Method: Focusing on structured input settings, we introduce quantum cohomology techniques to construct a quantum framework capable of discriminating and tracking homology classes; design Betti number and persistent Betti number estimation algorithms leveraging quantum phase estimation and amplitude estimation, achieving polynomial speedup under strengthened assumptions; and propose the first quantum algorithm for homology class testing and discrimination. Contributions: (1) An efficient Betti number estimation scheme circumventing the NP-hardness barrier; (2) A homology class discrimination protocol based on cohomology operator construction and quantum measurement; (3) A quantum homology tracking framework enabling dynamic TDA analysis. Our approach bridges quantum computation with algebraic topology, offering structured quantum advantages in topological inference.

Technology Category

Machine Learning: Quantum Machine LearningHumans and AI: Other Foundations of Human Computation & AIKnowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphsWeb Mining and Content Analysis: Web data quality in the era of algorithmically-generated contentResponsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Recently, the application of quantum computation to topological data analysis (TDA) has received increasing attention. In particular, several quantum algorithms have been proposed for estimating (normalized) Betti numbers, a central challenge in TDA. However, it was recently proven that estimating Betti numbers is an NP-hard problem, revealing a complexity-theoretic limitation to achieving a generic quantum advantage for this task. Motivated by this limitation and inspired by previous progress, we explore broader quantum approaches to TDA. First, we consider scenarios in which a simplicial complex is specified in a more informative form, enabling alternative quantum algorithms to estimate Betti numbers and persistent Betti numbers. We then move beyond Betti numbers and study the problem of testing the homology class of a given cycle, as well as distinguishing between homology classes. We also introduce cohomological techniques for these problems, along with a quantum algorithm. We then discuss their potential use in the testing and tracking of homology classes, which can be useful for TDA applications. Our results show that, despite the hardness of general Betti number estimation, quantum algorithms can still offer speed-ups in structured settings.

Problem

• Estimating Betti numbers in quantum topological data analysis
• Testing homology class of cycles in structured settings
• Tracking homology classes using cohomological quantum techniques

Innovation

• Quantum algorithms for Betti number estimation
• Testing homology classes with quantum methods
• Cohomological techniques for homology tracking