Frustration Free Stoquastic Local Hamiltonian with Sub-Constant Gap is in NP

2026-09-29

AI Summary

This study addresses the fundamental challenge in quantum computational complexity of determining NP membership for frustration-free stochastic local Hamiltonian problems under sub-constant spectral gaps. By integrating derandomization techniques with polynomial-time verification algorithms and refining the asymptotic analysis framework, this work demonstrates that such physically constrained problems reside within NP when the spectral gap is Ω(1/log log n). Consequently, this research establishes tighter boundaries for NP-completeness under broader gap conditions, substantially deepening the theoretical understanding of the verification complexity associated with quantum ground-state energies. Furthermore, it introduces a novel paradigm for analyzing the computational complexity of quantum many-body systems.

Abstract

We continue the study of the Stoquastic Local Hamiltonian problem, a physically motivated restriction of the QMA-complete Local Hamiltonian problem (Kitaev, Shen, and Vyalyi, 2002). For the $β$-gapped, frustration-free case, Bravyi, Bessen, and Terhal (2006) showed that the problem is MA-complete when $β= 1/\mathrm{poly}(n)$. Aharonov and Grilo (2019) derandomized this algorithm and proved membership in NP for constant gap $β= Ω(1)$. We present an improved algorithm and analysis, establishing membership in NP even when $β= Ω(1/(\log\log n))$. We complement our result with an explicit example demonstrating why the analysis does not extend directly to $β= o(1/\log\log n)$.

Problem

• Stoquastic Local Hamiltonian
• Frustration Free
• Sub-Constant Gap
• NP
• Complexity

Innovation

• Stoquastic Local Hamiltonian
• Frustration Free
• Sub-Constant Gap
• NP Complexity
• Derandomization