Quantum Lovász local lemma: Shearer’s bound is tight

Abstract

Lovász 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 [23,28,35]. A tight criterion under which the abstract version LLL (ALLL) holds was given by Shearer [44]. It turns out that Shearer's bound is generally not tight for variable version LLL (VLLL) [24]. Recently, Ambainis et al. [3] introduced a quantum version LLL (QLLL), which was then shown to be powerful for the 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. [34,39]. Our result also shows the tightness of Gilyén and Sattath's algorithm [18], and implies that the lattice gas partition function fully characterizes quantum satisfiability for almost all Hamiltonians with large enough qudits [39].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. This result might imply that it is possible to design an algorithm for CLLL which is still efficient beyond Shearer's bound.In applications of LLLs, the symmetric cases are most common, i.e., the events are with the same probability [15,16] and the Hamiltonians are with the same relative dimension [3,39]. We give the first lower bound on the gap between the symmetric VLLL and Shearer's bound. Our result can be viewed as a quantitative study on the separation between quantum and classical constraint satisfaction problems. Additionally, we obtain similar results for the symmetric CLLL. As an application, we give lower bounds on the critical thresholds of VLLL and CLLL for several of the most common lattices.Classical Lovász Local Lemma Lovász Local Lemma (or 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, and has numerous applications. Formally, given a set A of bad events in a probability space, LLL provides the condition under which P(∩ A∈A A) > 0. The dependency among events is usually characterized by the dependency graph. A dependency graph is an undirected graph G D = ([m], E D ) such that for any vertex i, A i is independent of {A j : j / ∈ Γ i ∪ {i}}, where Γ i stands for the set of neighbors of i in G D . In this setting, finding the conditions under which P(∩ A∈A A) > 0 is reduced to the following problem: given a graph G D , determine its abstract interior I(G D ) which is the set of vectors p such that P ∩ A∈A A > 0 for any event set A with dependency graph G D and probability vector p. Local solutions to this problem, including the first LLL proved in 1975 by Erdős and Lovász ...

The Mathematics of László Lovász

“…It appears again and again in multiple applications, ramifications, and forms. It is not possible to cover here all the applications in Ramsey theory (see Spencer [153]), extremal combinatorics (see Alon and Spencer [6]), number theory, and elsewhere (see, e.g., Ambainis et al [7], He et al [75], and Szegedy [155]). It was also discovered, see [146], that the Lovász Local Lemma closely relates to important results of Dobrushin in statistical physics [37].…”

New versions of Lovász Local Lemma and their applications

“…定理17 [27] Table 2 Summary of the critical thresholds for various lattices [34] Lattice Quantum Lower bound of the difference (between the classical and quantum thresholds) Square 0.1193 [10,69] 5.943 × 10 −8 Hexagonal 0.1547 [10] 1.211 × 10 −7…”

“…虽然我们已经知道变量版本局部引理紧的条件, 即定理 9, 但定 理 9 中的数学规划很难求解, 要直接计算晶格上变量版本的临界阈值是非常困难的. 2019 年, 何昆 等 [34] 考虑了晶格上的事件系统. 他们借助变量版本、量子版本以及 Lopsided 版本局部引理紧的条 件, 即定理 9, 14 和 7, 给出了事件系统与局域哈密尔顿量临界阈值之差的下界, 见表 2 [10, 34, 67∼69] .…”

Comparison of Two Convergence Criteria for the Variable-Assignment Lopsided Lovasz Local Lemma

“…We also remark on other related criteria for the variable-assignment LLL setting. For instance, [9,10] derive certain convergence conditions in terms of the bipartite graph H on vertex sets {1, . .…”