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) 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) 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.
In this thesis, I study how quantum computation and quantum communication can help large networks of computers to solve computational problems faster. I also connect the notion of locality between various models, including distributed quantum algorithms and online graph algorithms, and show that locality in online graph algorithms can be used to provide bounds for distributed quantum advantage. Towards proving distributed quantum advantage, I present the first family of LCL problems which shows a quantum advantage for distributed graph algorithms. Here LCLs, or Locally Checkable Labeling problems, are a family of graph problems described by a finite set of local constraints, and I mainly focus on them in this thesis. The models of distributed computing which this result separates are the classical LOCAL model and the quantum-LOCAL model. In both of these models, a network of computers exchanges messages in synchronous rounds, and the running time of the algorithm directly corresponds with locality, that is, how far the computers see in the network. In this thesis, I also introduce new models of computing that capture locality in stronger settings. Most notably, I introduce the online-LOCAL model which captures the locality of online graph algorithms, and show that it is, asymptotically, roughly as strong as the classical LOCAL model for solving LCL problems on rooted regular trees. I provide an online-LOCAL algorithm for finding a 3-coloring of a bipartite graph with lower locality than is possible in the classical LOCAL model, and show that the algorithm is tight. To draw things together, I show that quantum-LOCAL and online-LOCAL are indeed related to each other: any quantum-LOCAL algorithm can be simulated in the randomized online-LOCAL model. The converse does not hold in general, but an online-LOCAL algorithm solving an LCL problem in rooted, possibly non-regular, trees can be simulated in the classical LOCAL model—and thus in the quantum-LOCAL model—with an exponential cost in locality, putting indirectly a bound on the maximum possible attainable quantum advantage. Finally, I consider the sinkless orientation problem, which holds a central role in our understanding of the locality of LCL problems in the classical LOCAL model, and provide a simpler and more direct proof for its hardness using the supported LOCAL model. I also provide a simpler algorithm solving sinkless orientation in the SLOCAL model, which is a sequential version of the LOCAL model.
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.
We connect three distinct lines of research that have recently explored extensions of the classical LOCAL model of distributed computing: A. distributed quantum computing and non-signaling distributions [e.g. STOC 2024], B. finitely-dependent processes [e.g. Forum Math. Pi 2016], and C. locality in online graph algorithms and dynamic graph algorithms [e.g. ICALP 2023]. We prove new results on the capabilities and limitations of all of these models of computing, for locally checkable labeling problems (LCLs). We show that all these settings can be sandwiched between the classical LOCAL model and what we call the randomized online-LOCAL model. Our work implies limitations on the quantum advantage in the distributed setting, and we also exhibit a new barrier for proving tighter bounds. Our main technical results are these: 1. All LCL problems solvable with locality O(log⋆n) in the classical deterministic LOCAL model admit a finitely-dependent distribution with locality O(1). This answers an open question by Holroyd [2024], and also presents a new barrier for proving bounds on distributed quantum advantage using causality-based arguments. 2. In rooted trees, if we can solve an LCL problem with locality o(logloglogn) in the randomized online-LOCAL model (or any of the weaker models, such as quantum-LOCAL), we can solve it with locality O(log⋆n) in the classical deterministic LOCAL model. One of many implications is that in rooted trees, O(log⋆n) locality in quantum-LOCAL is not stronger than O(log⋆n) locality in classical LOCAL.
We study lower bounds for quantum distributed computing, where a set of nodes (representing quantum computers) interconnected by an underlying network consisting of (bandwidth-restricted) links, communicate using quantum communication. Nodes have unlimited computational power and may share an unlimited number of entangled qubits. Our main contribution is a simple uniform technique for proving lower bounds for quantum distributed algorithms. In particular, we identify a new quantum communication model called the Server model which provides a connection between distributed algorithms and communication complexity: it is strong enough to capture the hardness of several distributed computing problems, while weak enough that several hard problems in two-party communication complexity remain hard; to this end, we identify a set of communication complexity techniques that can be carried over to the Server model, namely techniques based on nonlocal games. We show that these techniques serve as a fundamental tool in proving lower bounds in quantum distributed computing. The new techniques help us to prove several non-trivial quantum distributed lower bounds (which are the first-known quantum bounds for fundamental global problems such as minimum spanning tree, shortest paths etc.), some of which are new even in the classical setting. First, it allows us to show that all previous classical lower bounds of more than twenty verification and optimization graph problems in [Das Sarma et al., STOC 11] also hold in the quantum setting. Many of these bounds are tight, implying a large class of problems that do not gain an advantage from quantum effects. Our results also imply the following new results in the classical communication models: (1) the first randomized lower bounds for Hamiltonian cycle and spanning tree verification problems in both the distributed computing and the communication complexity model, answering the open problem of Das Sarma et al. and subsuming many bounds in [Babai, Frankl, and Simon, FOCS 96], and (2) the first lower bound that is tight for all weight aspect ratios, matching previous upper bounds of [Elkin, STOC 04]. Submitted for a Regular Presentation. The full version of this paper can be found as [15] at http://arxiv.org/abs/1207.5211. ∗Department of Computer Science, Ben-Gurion University, Beer-Sheva, 84105, Israel. E-mail: elkinm@cs.bgu.ac.il. †Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 & Centre for Quantum Technologies, National University of Singapore, Singapore 117543. E-mail: hklauck@gmail.com. Research at the Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation. ‡Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371. Work partially done while at University of Vienna, Austria. E-mail: danupon@gmail.com. §Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 & Department of Computer Science, Brown University, Providence, RI 02912, USA. E-mail: gopalpandurangan@gmail.com. Supported in part by the following research grants: Nanyang Technological University grant M58110000, Singapore MOE Academic Research Fund (AcRF) Tier 2 grant MOE2010-T2-2-082, and a grant from the US-Israeli Binational Science Foundation (BSF).
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.