发表机构
Nagoya University; Paderborn University; University of Hyogo（名古屋大学; 帕德博恩大学; 兵库大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究在无挫折和近似正则假设下，为局部哈密顿量问题提出指数加速的经典与量子算法，并证明量子5-SAT在(Q)SETH下无最坏情况加速，确立了度平衡作为细粒度复杂性边界。
AI 中文摘要
局部哈密顿量问题是典型的$\mathsf{QMA}$-完全问题，已知最坏情况下经典算法需要$O(2^n)$时间，量子算法需要$O(2^{n/2})$时间。对于广泛的问题类别，如何改进这些暴力策略尚不清楚，因为基态通常高度纠缠，无法直接应用经典CSP的已知策略。在本工作中，我们在两个温和假设下提出了指数加速的经典和量子算法：（1）哈密顿量在YES实例上是无挫折的，（2）它是近似正则的，意味着每个量子比特大约受到相同数量的约束作用。我们通过证明在(Q)SETH假设下，量子5-SAT不存在非平凡的最坏情况加速来补充这些上界。我们的下界进一步表明，我们的算法对正则性的依赖在某种意义上是近乎最优的。具体而言，量子5-SAT即使在所有但$O(\sqrt{n})$个量子比特仅参与常数个约束，而剩余的$O(\sqrt{n})$个量子比特每个参与$O(\sqrt{n})$个约束的哈密顿量上仍然是(Q)SETH难的。相比之下，如果这个高度子集的大小或其量子比特的度数减少$n^\delta$倍（对于任何$\delta>0$），我们的算法在$O(2^{(1-\varepsilon)n})$时间内解决问题，其中$\varepsilon>0$。综上，我们的上下界为量子可满足性建立了细粒度复杂性二分法。
英文摘要
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.
Comments40 pages
