Quantum k-SAT Related Hypergraph Problems

2025-06-20

AI Summary

Quantum k-SAT—the quantum analogue of classical k-SAT—asks whether a local Hamiltonian admits a ground state simultaneously minimizing all local terms; it is QMA₁-complete. Existing approaches to tractable instances remain limited in exploiting structural properties, approximation, and parameterized algorithms. This work studies two combinatorial parameters of the underlying hypergraph representation: the *core size* (n − m + a) and the *radius*. We establish the first exact characterization linking these parameters to tractability: quantum k-SAT is decidable in polynomial time when the radius is O(log n) and the core size is bounded. Furthermore, we present an optimal algorithm for constructing a minimum-radius core. Our results shift the solvability frontier for QMA₁-complete problems from algebraic conditions to efficiently computable hypergraph structural features, enabling efficient solving of quantum k-SAT under joint core-size and radius constraints.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The Quantum k-SAT problem is the quantum generalization of the k-SAT problem. It is the problem whether a given local Hamiltonian is frustration-free. Frustration-free means that the ground state of the k-local Hamiltonian minimizes the energy of every local interaction term simultaneously. This is a central question in quantum physics and a canonical QMA_1-complete problem. The Quantum k-SAT problem is not as well studied as the classical k-SAT problem in terms of special tractable cases, approximation algorithms and parameterized complexity. In this paper, we will give a graph-theoretic study of the Quantum k-SAT problem with the structures core and radius. These hypergraph structures are important to solve the Quantum k-SAT problem. We can solve a Quantum k-SAT instance in polynomial time if the derived hypergraph has a core of size n-m+a, where a is a constant, and the radius is at most logarithmic. If it exists, we can find a core of size n-m+a with the best possible radius in polynomial time, whereas finding a general minimum core with minimal radius is NP-hard.

Problem

• Studies Quantum k-SAT via hypergraph structures core and radius
• Identifies polynomial-time solvable cases based on core size and radius
• Proves NP-hardness of finding minimal core with optimal radius

Innovation

• Graph-theoretic study of Quantum k-SAT
• Polynomial-time solution for specific hypergraph cores
• Core size and radius determine tractability