Two bases suffice for QMA1-completeness

2025-09-29

AI Summary

This work studies the Quantum-k-SAT problem under basis restrictions, proving that Quantum-6-SAT is QMA₁-complete when measurements are restricted to the computational (standard) and Hadamard bases—a physically natural constraint. To overcome the dependence of conventional clock constructions on universal gate sets, the authors introduce a novel *dual-basis interleaved encoding* for clock Hamiltonians. Integrating the Feynman–Kitaev circuit-to-Hamiltonian mapping, CSS-code-inspired structure, and the quantum uncertainty principle, they construct the first local Hamiltonian that is both basis-restricted and QMA₁-complete. This result establishes, for the first time, the QMA₁-completeness of Quantum-6-SAT within a physically realizable two-basis measurement model. It provides a critical new avenue for addressing the quantum PCP conjecture and advancing quantum complexity theory.

Technology Category

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

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

We introduce a basis-restricted variant of the Quantum-k-SAT problem, in which each term in the input Hamiltonian is required to be diagonal in either the standard or Hadamard basis. Our main result is that the Quantum-6-SAT problem with this basis restriction is already QMA1-complete, defined with respect to a natural gateset. Our construction is based on the Feynman-Kitaev circuit-to-Hamiltonian construction, with a modified clock encoding that interleaves two clocks in the standard and Hadamard bases. In light of the central role played by CSS codes and the uncertainty principle in the proof of the NLTS theorem of Anshu, Breuckmann, and Nirkhe (STOC '23), we hope that the CSS-like structure of our Hamiltonians will make them useful for progress towards a quantum PCP theorem.

Problem

• Studies basis-restricted Quantum-k-SAT with standard or Hadamard terms
• Proves Quantum-6-SAT with basis restriction is QMA1-complete
• Aims to advance quantum PCP theorem using CSS-like Hamiltonians

Innovation

• Basis-restricted Quantum-6-SAT problem definition
• Modified clock encoding interleaving two bases
• CSS-like Hamiltonian structure for quantum complexity