Superlogarithmic Gap Result for LCLs on Trees in Quantum-LOCAL

2026-08-17

AI Summary

This study investigates complexity gaps for Locally Checkable Labeling (LCL) problems on trees within the quantum LOCAL model by integrating rake-and-compress tree decomposition with bounded-dependence distributional local simulation. The core contribution establishes that any LCL problem solvable via low-dependence distributions admits a deterministic logarithmic-time algorithm, thereby confirming a super-logarithmic complexity gap in this setting. This result demonstrates a strict dichotomy wherein such problems either possess efficient classical solutions or require polynomially many quantum rounds. Consequently, this finding provides a critical theoretical foundation for characterizing quantum advantages in distributed computing, precisely delineating the boundary between classical tractability and genuine quantum speedup for LCL problems on tree topologies within the quantum LOCAL framework.

Abstract

We show that, on trees, any locally checkable labeling problem (LCL) $Π$ that can be solved by an $n^{o(1)}$-dependent distribution can also be solved by an $O(\log n)$-round deterministic LOCAL algorithm. The result is obtained through a rake-and-compress-style decomposition of the input tree, and local simulations of the bounded dependent distribution on the components of the decomposition. As a corollary to our result, any LCL problem on trees can either be solved by an $O(\log n)$ deterministic LOCAL algorithm, or requires $n^{Ω(1)}$ rounds to solve by a quantum-LOCAL algorithm.

Problem

• Quantum-LOCAL
• Locally Checkable Labeling
• Trees
• Complexity Gap
• Distributed Computing

Innovation

• Quantum-LOCAL
• Locally Checkable Labeling
• Rake-and-Compress Decomposition
• Superlogarithmic Gap
• Deterministic LOCAL Algorithm