Quantum Lov{á}sz Local Lemma: Shearer's Bound is Tight

Lov{\'a}sz 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" condition. Over the last decades, the algorithmic aspect of LLL has also attracted lots of attention in theoretical computer science ~\cite{moser2010constructive,kolipaka2011moser,harvey2015algorithmic}. A tight criterion under which the \emph{abstract} version LLL (ALLL) holds was given by Shearer ~\cite{shearer1985problem}. It turns out that Shearer's bound is generally not tight for \emph{variable} version LLL (VLLL)~\cite{he2017variable}. Recently, Ambainis et al. \cite{ambainis2012quantum} introduced a quantum version LLL (QLLL), which was then shown to be powerful for quantum satisfiability problem. In this paper, 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, affirming a conjecture proposed by Sattath et al.~\cite{pnas,Morampudi2018Many}. Our result also shows the tightness of Gily{\'e}n and Sattath's algorithm \cite{gilyen2016preparing}, and implies that the lattice gas partition function fully characterizes quantum satisfiability for almost all Hamiltonians with large enough qudits~\cite{pnas}. Commuting LLL (CLLL), LLL for commuting local Hamiltonians which are widely studied in the literature, is also investigated here. We prove that the tight regions of CLLL and QLLL are different in general. Thus, the efficient region of algorithms for CLLL can go beyond Shearer's bound. In applications of LLLs, the symmetric cases are most common, i.e., the events are with the same probability ~\cite{gebauer2016local,gebauer2009lovasz} and the Hamiltonians are with the same relative dimension ~\cite{ambainis2012quantum,pnas}. [See pdf file for full abstract]