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arXiv:1804.07055 · ACM Symposium on Theory of Computing (STOC)

Quantum Lovász local lemma: Shearer’s bound is tight

Kun He, Qian Li, Xiaoming Sun, Jiapeng Zhang

Published Jun 20, 2019• 4 citations
arXivHTMLPDF

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. A tight criterion under which the abstract version LLL (ALLL) holds was given by Shearer. It turns out that Shearer’s bound is generally not tight for variable version LLL (VLLL). Recently, Ambainis et al. introduced a quantum version LLL (QLLL), which was then shown to be powerful for the quantum satisfiability problem.

DOI
10.1145/3313276.3316392

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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. A tight criterion under which the abstract version LLL (ALLL) holds was given by Shearer. It turns out that Shearer’s bound is generally not tight for variable version LLL (VLLL). Recently, Ambainis et al. introduced a quantum version LLL (QLLL), which was then shown to be powerful for the quantum satisfiability problem.
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 (quantum analog of k - sat ): For instance we show that any collection of rank-1 projectors, with the property that each qubit appears in at most 2 k /( e ċ 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. [2009, 2010] 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.
The recently proven Quantum Lovasz Local Lemma generalises the well-known Lovasz Local Lemma. It states that, if a collection of subspace constraints are "weakly dependent", there necessarily exists a state satisfying all constraints. It implies e.g. that certain instances of the kQSAT quantum satisfiability problem are necessarily satisfiable, or that many-body systems with "not too many" interactions are always frustration-free. However, the QLLL only asserts existence; it says nothing about how to find the state. Inspired by Moser's breakthrough classical results, we present a constructive version of the QLLL in the setting of commuting constraints, proving that a simple quantum algorithm converges efficiently to the required state. In fact, we provide two different proofs, one using a novel quantum coupling argument, the other a more explicit combinatorial analysis. Both proofs are independent of the QLLL. So these results also provide independent, constructive proofs of the commutative QLLL itself, but strengthen it significantly by giving an efficient algorithm for finding the state whose existence is asserted by the QLLL. We give an application of the constructive commutative QLLL to convergence of CP maps. We also extend these results to the non-commutative setting. However, our proof of the general constructive QLLL relies on a conjecture which we are only able to prove in special cases.
In this work, we study the Lovász local lemma (LLL) problem in the area of distributed quantum computing, which has been the focus of attention of recent advances in quantum computing [STOC’24, STOC’25, STOC’25]. We prove a lower bound of 2Ω⁡(log∗⁡𝑛) for the complexity of the distributed LLL in the quantum-LOCAL model. More specifically, we obtain our lower bound already for a very well-studied special case of the LLL, called sinkless orientation, in a stronger model than quantum-LOCAL, called the randomized online-LOCAL model. As a consequence, we obtain the same lower bounds for sinkless orientation and the distributed LLL also in a variety of other models studied across different research communities. Our work provides the first superconstant lower bound for sinkless orientation and the distributed LLL in all of these models, addressing recently stated open questions. Moreover, to obtain our results, we develop an entirely new lower bound technique that we believe has the potential to become the first generic technique for proving post-quantum lower bounds for many of the most important problems studied in the context of locality.
In this work, we study the Lovász local lemma (LLL) problem in the area of distributed quantum computing, which has been the focus of attention of recent advances in quantum computing [STOC’24, STOC’25, STOC’25]. We prove a lower bound of 2Ω(log⁡∗n)2^{\Omega(\log^*n)}2Ω(log∗n) for the complexity of the distributed LLL in the quantum-LOCAL model. More specifically, we obtain our lower bound already for a very well-studied special case of the LLL, called sinkless orientation, in a stronger model than quantum-LOCAL, called the randomized online-LOCAL model. As a consequence, we obtain the same lower bounds for sinkless orientation and the distributed LLL also in a variety of other models studied across different research communities.
A frustration-free local Hamiltonian has the property that its ground state minimises the energy of all local terms simultaneously. In general, even deciding whether a Hamiltonian is frustration-free is a hard task, as it is closely related to the QMA1-complete quantum satisfiability problem (QSAT) - the quantum analogue of SAT, which is the archetypal NP-complete problem in classical computer science. This connection shows that the frustration-free property is not only relevant to physics but also to computer science. The Quantum Lovasz Local Lemma (QLLL) provides a sufficient condition for frustration-freeness. Is there an efficient way to prepare a frustration-free state under the conditions of the QLLL? Previous results showed that the answer is positive if all local terms commute. These works were based on Moser's “compression argument” which was the original analysis technique of the celebrated resampling algorithm. We generalise and simplify the “compression argument”, so that it provides a simplified version of the previous quantum results, and improves on some classical results as well. More importantly, we improve on the previous constructive results by designing an algorithm that works efficiently for non-commuting terms as well, assuming that the system is “uniformly” gapped, by which we mean that the system and all its subsystems have an inverse polynomial energy gap. Similarly to the previous results, our algorithm has the charming feature that it uses only local measurement operations corresponding to the local Hamiltonian terms.