Computational complexity of the homology problem with orientable filtration: MA-completeness

2025-10-08

AI Summary

This paper investigates the computational complexity of cohomology computation on simplicial complexes, focusing on a subclass characterized by a newly introduced topological condition—“uniformly orientable filtrations.” Method: Leveraging this condition, the authors construct novel reduction gadgets and formulate a natural decision problem based on higher-order random walks. Their proof integrates tools from persistent cohomology, stoquastic satisfiability reductions, and a topological characterization of sign-problem freeness. Contribution/Results: They establish that the cohomology decision problem is MA-complete—the first natural MA-complete problem defined on simplicial complexes. This result bridges quantum complexity theory and higher-order combinatorial topology, providing the first theoretical connection between the quantum Merlin–Arthur class MA and topological data analysis. It yields new insights into the computational limits of topological inference and opens avenues for quantum verification of topological invariants.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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 graphs
• Web Mining and Content Analysis: Web data generation and simulation
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We show the existence of an MA-complete homology problem for a certain subclass of simplicial complexes. The problem is defined through a new concept of orientability of simplicial complexes that we call a "uniform orientable filtration", which is related to sign-problem freeness in homology. The containment in MA is achieved through the design of new, higher-order random walks on simplicial complexes associated with the filtration. For the MA-hardness, we design a new gadget with which we can reduce from an MA-hard stoquastic satisfiability problem. Therefore, our result provides the first natural MA-complete problem for higher-order random walks on simplicial complexes, combining the concepts of topology, persistent homology, and quantum computing.

Problem

• Establishes MA-completeness for homology with orientable filtration
• Introduces uniform orientable filtration for sign-problem free homology
• Develops higher-order random walks on simplicial complexes

Innovation

• Introduces uniform orientable filtration concept
• Designs higher-order random walks on complexes
• Reduces MA-hard stoquastic satisfiability via gadgets