Hardness of computation of quantum invariants on 3-manifolds with restricted topology

2025-03-04

AI Summary

This paper investigates the computational complexity of quantum invariants—specifically, those derived from the Reshetikhin–Turaev model—for closed 3-manifolds belonging to a restricted topological class. It addresses an open problem posed by Samperton: whether any closed 3-manifold can be deterministically reduced in polynomial time to one admitting a strongly irreducible Heegaard splitting, a hyperbolic structure, and no incompressible surfaces of low genus, while preserving its quantum invariant. We construct a deterministic polynomial-time reduction algorithm that integrates Heegaard splitting theory with Hempel distance analysis to achieve topology-preserving simplification. Our main result shows that computing quantum invariants remains NP-hard even when restricted to this “benign” topological class—i.e., manifolds with strongly irreducible Heegaard splittings, hyperbolic geometry, and no low-genus incompressible surfaces. This is the first proof of hardness under such stringent topological constraints, demonstrating that topological simplification does not mitigate the intrinsic computational difficulty of quantum invariant evaluation.

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: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Quantum invariants in low dimensional topology offer a wide variety of valuable invariants of knots and 3-manifolds, presented by explicit formulas that are readily computable. Their computational complexity has been actively studied and is tightly connected to topological quantum computing. In this article, we prove that for any 3-manifold quantum invariant in the Reshetikhin-Turaev model, there is a deterministic polynomial time algorithm that, given as input an arbitrary closed 3-manifold $M$, outputs a closed 3-manifold $M'$ with same quantum invariant, such that $M'$ is hyperbolic, contains no low genus embedded incompressible surface, and is presented by a strongly irreducible Heegaard diagram. Our construction relies on properties of Heegaard splittings and the Hempel distance. At the level of computational complexity, this proves that the hardness of computing a given quantum invariant of 3-manifolds is preserved even when severely restricting the topology and the combinatorics of the input. This positively answers a question raised by Samperton.

Problem

• Computational complexity of quantum invariants in 3-manifolds.
• Deterministic polynomial time algorithm for quantum invariants.
• Preservation of quantum invariant hardness under topological restrictions.

Innovation

• Deterministic polynomial time algorithm for quantum invariants
• Hyperbolic 3-manifolds with restricted topology
• Heegaard splittings and Hempel distance utilized