The Lov\'asz Local Lemma (LLL) is a very powerful tool in combinatorics and probability theory to show the possibility of avoiding all bad events under some weakly dependent conditions. In a seminal paper, Ambainis, Kempe, and Sattath (JACM 2012) introduced a quantum version LLL (QLLL) which shows the possibility of avoiding all ``bad" Hamiltonians under some weakly dependent condition, and applied QLLL to the random k-QSAT problem. Sattath, Morampudi, Laumann, and Moessner (PNAS 2015) extended Ambainis, Kempe, and Sattath's result and showed that Shearer's bound is a sufficient condition for QLLL, and conjectured that Shearer's bound is indeed the tight condition for QLLL. In this paper, we affirm this conjecture. Precisely, we prove that Shearer's bound is tight for QLLL, i.e., the relative dimension of the smallest satisfying subspace is completely characterized by the independent set polynomial. Our result implies the tightness of Gily\'en and Sattath's algorithm (FOCS 2017), and also implies that the lattice gas partition function fully characterizes quantum satisfiability for almost all Hamiltonians with large enough qudits (Sattath, Morampudi, Laumann and Moessner, PNAS 2015). The commuting LLL (CLLL), which focuses on commuting local Hamiltonians, is also investigated here. We prove that the tight regions of CLLL and QLLL are different in general. This result indicates that it is possible to design an algorithm for CLLL which is still efficient beyond Shearer's bound.

翻译：Lovász局部引理（LLL）是组合数学和概率论中一个非常强大的工具，用于证明在某些弱依赖条件下避免所有坏事件的可能性。在一篇开创性论文中，Ambainis、Kempe和Sattath（JACM 2012）引入了量子版本LLL（QLLL），该引理表明在某些弱依赖条件下避免所有“坏”哈密顿量的可能性，并将QLLL应用于随机k-QSAT问题。Sattath、Morampudi、Laumann和Moessner（PNAS 2015）扩展了Ambainis、Kempe和Sattath的结果，证明Shearer界是QLLL的充分条件，并推测Shearer界实际上是QLLL的紧条件。本文证实了这一猜想。具体而言，我们证明了Shearer界对于QLLL是紧的，即最小满足子空间的相对维度完全由独立集多项式刻画。我们的结果暗示了Gilyén和Sattath算法（FOCS 2017）的紧性，同时也表明晶格气体配分函数完全刻画了几乎所有具有足够大qudit的哈密顿量的量子可满足性（Sattath、Morampudi、Laumann和Moessner，PNAS 2015）。本文也研究了关注对易局部哈密顿量的对易LLL（CLLL）。我们证明CLLL和QLLL的紧区域在一般情况下是不同的。这一结果表明，有可能为CLLL设计一种在Shearer界之外仍然高效的算法。