Quantum Lov\'asz Local Lemma: Shearer's Bound is Tight

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.