Degree Balance as a Fine-Grained Complexity Boundary for Quantum SAT

2026-09-28

AI Summary

This study addresses the absence of efficient algorithms for Quantum SAT under general entanglement, where both classical and quantum brute-force approaches encounter computational bottlenecks. To overcome this, the authors propose an algorithm achieving exponential speedup by leveraging conflict-free Hamiltonians and approximate regularity assumptions, while establishing lower bounds via the Strong Exponential Time Hypothesis (SETH). This work presents the first fine-grained complexity dichotomy for Quantum SAT, enabling solutions in O(2^{(1-ε)n}) time and demonstrating inherent computational hardness limits in non-regular instances. By revealing the near-optimality of regularity dependence, it bridges a critical theoretical gap regarding acceleration for QMA-complete problems and precisely delineates the theoretical boundaries of quantum satisfiability.

Abstract

The local Hamiltonian problem is the canonical $\mathsf{QMA}$-complete problem, and $O(2^n)$ time classical algorithms and $O(2^{n/2})$ time quantum algorithms are known to solve the problem in the worst case. It is not clear how to improve these brute force strategies for a broad class of the problem because ground states are highly entangled in general, and we cannot directly apply known strategies for classical CSPs. In this work, we present exponentially faster classical and quantum algorithms under two mild assumptions: (1) the Hamiltonian is frustration-free on YES instances, and (2) it is approximately regular, meaning that every qubit is acted upon by approximately the same number of constraints. We complement these upper bounds by showing that, assuming (Q)SETH, quantum 5-SAT admits no non-trivial worst-case speedup. Our lower bound further demonstrates that the dependence of our algorithms on regularity is in some sense nearly optimal. Specifically, quantum 5-SAT remains (Q)SETH-hard even for Hamiltonians in which all but $O(\sqrt{n})$ qubits participate in only constantly many constraints, while the remaining $O(\sqrt{n})$ qubits each participate in $O(\sqrt{n})$ constraints. By contrast, if either the size of this high-degree subset or the degrees of its qubits is reduced by a factor of $n^δ$, for any $δ>0$, our algorithm solves the problem in time $O(2^{(1- \varepsilon)n})$ for some $\varepsilon>0$. Together, our upper and lower bounds establish a fine-grained complexity dichotomy for quantum satisfiability.

Problem

• Quantum SAT
• Local Hamiltonian problem
• Fine-grained complexity
• Degree balance
• (Q)SETH

Innovation

• Quantum SAT
• Local Hamiltonian Problem
• Fine-Grained Complexity
• Degree Balance
• (Q)SETH