A Post-Quantum Lower Bound for the Distributed Lovász Local Lemma

TL;DR

The paper tackles post-quantum lower bounds for distributed Lovász Local Lemma problems by introducing the randomized online-LOCAL model and a novel construction-tree based method. It proves a new superconstant lower bound of 2^{Ω(log^* n)} for sinkless orientation and the distributed LLL, already on degree-4 trees, and shows that this bound transfers to models at least as weak as randomized online-LOCAL, including quantum-LOCAL. The approach relies on a generic technique built from reflection and split operations, encoded in a construction tree, with a lifting mechanism that doubles the complexity bound at each transformation. These results address open questions about post-quantum locality and pave the way for a broader, generic lower-bound framework applicable to other locally checkable problems in quantum-augmented distributed settings.

Abstract

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 $2^{Ω(\log^* n)}$ 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.

Figures (3)

• Figure 1: A high-level overview of our approach in the deterministic online-LOCAL model for the case $L = 2$ and $\Delta = 3$. Each box displays a configuration as discussed in the high-level overview in \ref{['sec:approach']}. A downward arrow between configuration corresponds to a reflection operation, an upward arrow to a split operation, where in the source configuration the node to be removed is indicated by a dotted circle and in the target configuration only one of the isomorphic connected subconfigurations of the resulting configuration is displayed. The final (small) box represents the empty configuration.
• Figure 2: A slightly simplified overview of the input graph and query sequence obtained from the approach illustrated in \ref{['fig:initialexample']} for $L = 2$ (i.e., considering an algorithm $\mathcal{A}$ with complexity $L - 1 = 1$) and $\Delta = 3$. The solid nodes represent the first nodes in the query sequence, in the order displayed below the nodes. For each solid node one incident edge is displayed as being oriented away from the node, illustrating the part of the decision of $\mathcal{A}$ for the respective node that is relevant for obtaining the desired contradiction. Note that each time $\mathcal{A}$ has to make a decision for a solid node $v$, the situation regarding the connected component containing $v$ of the subgraph that $\mathcal{A}$ has already seen at this point in time is symmetric, essentially allowing us to choose which incident edge we want $\mathcal{A}$ to orient outwards. With the $14$th presented node, $\mathcal{A}$ will create an incorrect output. We remark that, unfortunately, the small examples given in this figure and \ref{['fig:initialexample']} are misleading in their simplicity; some assumptions that one is tempted to make by looking at the figures do actually not hold when considering larger $L$. For instance, the aforementioned symmetry does not hold in general; however it holds if one ignores the parts of the connected component that are "behind" a node that is oriented towards the considered node $v$. We also note that the aforementioned "slight simplification" of \ref{['fig:initialqueryseq']} merely consists in omitting some nodes that are incident to the displayed nodes and guarantee that every displayed node has degree $\Delta = 3$ (which is relevant due to the precise definition of the deterministic online-LOCAL model, in particular regarding which information the algorithm receives).
• Figure 3: A $b$-ary construction tree for $b = 3$. Circles and diamonds represent reflect nodes and split nodes, respectively. The string inside a circle/diamond is the label of the respective node.

Theorems & Definitions (53)

• Lemma 1.1: LLL
• Theorem 1.2
• Corollary 1.3
• Definition 1.4: seeing a node/edge
• Definition 2.1
• Definition 2.2: $b$-balanced tree
• Definition 2.3: well-nested
• Lemma 2.4
• proof
• Definition 2.5: $j$-th reflect/split layer