Distributed Quantum Advantage in Locally Checkable Labeling Problems

2025-04-07

AI Summary

Whether local checkable labeling (LCL) problems admit distributed quantum advantage in the LOCAL model. Method: The authors construct the first explicit LCL problem solvable in $O(log n)$ rounds in the quantum-LOCAL model, while requiring $Omega(log n cdot log^{0.99} log n)$ rounds for any classical randomized LOCAL algorithm. They further establish a general upper bound via a classical simulation of quantum protocols using random walks and amplitude estimation. Results: This yields the first asymptotic quantum advantage for LCL problems. Moreover, they prove that any LCL problem solvable in $T(n)$ rounds by a quantum-LOCAL algorithm is also solvable classically in $widetilde{O}(sqrt{n T(n)})$ rounds—thereby characterizing the fundamental limits of quantum speedup for LCLs. Crucially, this bound rules out the existence of finite-dependence distributions for global LCL problems, providing a pivotal complexity-theoretic barrier for quantum distributed computing.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

In this paper, we present the first known example of a locally checkable labeling problem (LCL) that admits asymptotic distributed quantum advantage in the LOCAL model of distributed computing: our problem can be solved in $O(log n)$ communication rounds in the quantum-LOCAL model, but it requires $Omega(log n cdot log^{0.99} log n)$ communication rounds in the classical randomized-LOCAL model. We also show that distributed quantum advantage cannot be arbitrarily large: if an LCL problem can be solved in $T(n)$ rounds in the quantum-LOCAL model, it can also be solved in $ ilde O(sqrt{n T(n)})$ rounds in the classical randomized-LOCAL model. In particular, a problem that is strictly global classically is also almost-global in quantum-LOCAL. Our second result also holds for $T(n)$-dependent probability distributions. As a corollary, if there exists a finitely dependent distribution over valid labelings of some LCL problem $Pi$, then the same problem $Pi$ can also be solved in $ ilde O(sqrt{n})$ rounds in the classical randomized-LOCAL and deterministic-LOCAL models. That is, finitely dependent distributions cannot exist for global LCL problems.

Problem

• Demonstrates distributed quantum advantage in LCL problems
• Establishes limits on quantum advantage in LOCAL model
• Links finitely dependent distributions to classical solvability

Innovation

• Quantum-LOCAL model solves LCL in O(log n) rounds
• Classical-LOCAL requires Ω(log n · log^0.99 log n) rounds
• Quantum advantage bounded by classical O(√(n T(n)))