A quantum lovász local lemma

Abstract

The Lovász Local Lemma (LLL) is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of "weakly dependent" criteria. We show that the LLL extends to a much more general geometric setting, where events are replaced with subspaces and probability is replaced with relative dimension, which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions.Our result immediately applies to the k-QSAT problem: For instance we show that any collection of rank 1 projectors with the property that each qubit appears in at most 2 k /(e · k) of them, has a joint satisfiable state.We then apply our results to the recently studied model of random k-QSAT. Recent works have shown that the satisfiable region extends up to a density of 1 in the large k limit, where the density is the ratio of projectors to qubits. Using a hybrid approach building on work by Laumann et al. [LLM + 09] we greatly extend the known satisfiable region for random k-QSAT to a density of Ω(2 k /k 2 ). Since our tool allows us to show the existence of joint satisfying states without the need to construct them, we are able to penetrate into regions where the satisfying states are conjectured to be entangled, avoiding the need to construct them, which has limited previous approaches to product states.

Product, generic, and random generic quantum satisfiability

“…IV, we address the situation for larger values of α by adducing some primarily numerical evidence about the existence of random instances which are SAT but not PRODSAT. We note that recently, Ambainis et al [8] have proven the existence of such an entangled SAT phase for k ≥ 12 using their newly proven quantum Lovász local lemma coupled with the characterizations presented in this paper. We will have more to say about this exciting development in the conclusion.…”

“…Ambainis and co-workers [8] had recently proven a quantum version of the Lovász local lemma, which directly shows the quantum satisfiability of graphs G of bounded degree. Due to this degree restriction, the lemma does not directly apply to the Erdös random graph ensemble.…”

Quantum 3-SAT Is QMA1-Complete

“…As in the classical case, it is conjectured that a satisfiability threshold α c (k) exists, above which the probability that a random instance is satisfiable approaches zero as n → ∞ and below which this probability approaches one [16]. Some bounds on this threshold value have been proven using a quantum version of the Lovász local lemma [2] and by using graph-theoretic techniques [6] but only the case k = 2 is fully understood [16], [10]. Other previous work has focused on quantum satisfiability with qudit variables of dimension d > 2 [18], [20], [8], [5] or in restricted geometries [18], [7].…”

Quantum Lovász local lemma: Shearer’s bound is tight

“…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.…”