Online Locality Meets Distributed Quantum Computing

TL;DR

This work connects three distinct lines of research that have recently explored extensions of the classical LOCAL model of distributed computing, and proves new results on the capabilities and limitations of all of these models of computing, for locally checkable labeling problems (LCLs).

Abstract

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 $O(\log^\star n)$ 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 $o(\log \log \log n)$ in the randomized online-LOCAL model (or any of the weaker models, such as quantum-LOCAL), we can solve it with locality $O(\log^\star n)$ in the classical deterministic LOCAL model. One of many implications is that in rooted trees, $O(\log^\star n)$ locality in quantum-LOCAL is not stronger than $O(\log^\star n)$ locality in classical LOCAL.

Figures (10)

• Figure 1: A decomposition of rooted pseudoforests in directed paths and cycles: each node $v$ colors its in-neighbors with a uniformly sampled permutation of the elements of $[\text{indeg}(v)]$. The graph induced by nodes colored with color $i$ is a disjoint union of directed paths and cycles.
• Figure 2: A decomposition of a graph of maximum degree $\Delta = 5$ in rooted pseudoforests: for the sake of image clarity, we focus on the undirected case. In \ref{['fig:fin-dep:bounded-graphs-1']}, each node $v$ rearranges its port numbers with a uniformly sampled permutation of the elements of $[\deg(v)]$. As shown in \ref{['fig:fin-dep:bounded-graphs-2']}, edges hosting port number $i$ at some endpoint are oriented away from that port (in case both endpoints host port number $i$, the edge is duplicated) and form a rooted pseudoforest.
• Figure 3: A $(3,4)$-clustering of a rooted tree. The leader nodes are colored and their closed clusters marked with their respective color.
• Figure 4: Example of a $3$-colored grid with two boundaries $B_1$ and $B_2$ that have different parities. As one can see, $b(u,v) = 2$: No matter if we take the blue or the red path from $u$ to $v$, we always cross $B_1$ and then $B_2$.
• Figure 5: How to turn paths with large $b$-value into a contradiction. Here the blue area includes the nodes revealed so far around the path segments $(u_0,\ldots,u_L)$ and the corresponding part of $P_2$ underneath it. The green segments are the two segments we consider in the proof. A cycle going through both paths leads to a contradiction.

Theorems & Definitions (77)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 3.1
• definition 1: Labeling problem
• definition 2: Locally checkable labeling problem
• definition 3: Outcome
• definition 4: Non-signaling outcome