No Distributed Quantum Advantage for 3-Coloring Rooted Trees and 2-Coloring Even Cycles

Abstract: Significant effort has been devoted over the past decade to understanding whether quantum resources can provide advantages in distributed computing, and in particular whether they can help overcome locality constraints in networks, typically in Linial's LOCAL model. Recently, Coiteux-Roy~et~al.~(STOC 2024) showed that quantum resources do not help for 3-coloring \textit{unrooted} trees: in particular, their lower bound holds in the stronger \textit{non-signaling} model, which formalizes the principle of physical causality in distributed computing. The case of \textit{rooted} trees, however, was left open by their work. For rooted trees, the deterministic Cole-Vishkin algorithm 3-colors $n$-node trees in $O(\log^\star n)$ rounds, matching Linial's classical $Ω(\log^\star n)$ lower bound (FOCS 1987). In this paper, we show that any algorithm in quantum-LOCAL (without pre-shared entanglement) that properly 3-colors $n$-node rooted trees with probability at least ${1-O(1/\log n)}$ must perform $Ω(\log^\star n)$ rounds. That is, quantum resources provide no advantage for 3-coloring rooted trees. To get this result, we show a lower bound of $Ω(\log^\star Δ)$ for 3-coloring any $Δ$-ary tree with success probability at least $1-1/Δ$. The proof uses a \textit{color lifting} technique that bears similarity to Linial's original argument. We also show, as a separate result, that 2-coloring even-length $n$-node cycles with probability $1-O(1/n)$ requires $n/2-1$ rounds in the quantum-LOCAL model, even with pre-shared entangled states. This improves the previously known $\lceil (n-2)/4 \rceil$ lower bound of Gavoille, Kosowski, and Markiewicz (DISC 2009) by a factor of two, and shows that quantum algorithms cannot save even a single round over classical deterministic algorithms for 2-coloring even-length cycles.

• The paper establishes that any quantum-LOCAL algorithm for proper 3-coloring of rooted trees requires Ω(log* n) rounds, matching classical lower bounds.
• It proves that achieving 2-coloring for even cycles needs nearly n/2 rounds, thereby eliminating any potential quantum speedup.
• A novel color lifting technique and analysis of non-signaling output distributions extend classical round-elimination methods to the quantum domain.

No Distributed Quantum Advantage for 3-Coloring Rooted Trees and 2-Coloring Even Cycles

Introduction and Context

The paper "No Distributed Quantum Advantage for 3-Coloring Rooted Trees and 2-Coloring Even Cycles" (2607.04852) rigorously investigates the scope for quantum speed-up in distributed graph coloring tasks within the quantum-LOCAL model. It focuses on two canonical problems: (i) 3-coloring rooted trees, a fundamental primitive in the design of distributed algorithms, and (ii) 2-coloring even-length cycles, whose round complexity establishes the extreme locality barrier in classical distributed algorithms. The study is motivated by recent progress in distributed quantum algorithms, especially regarding the possibility that quantum correlations or pre-shared entanglement might circumvent established locality bounds, notably Linial’s $\Omega(\log^\star n)$ and $\Omega(n)$ lower bounds for these coloring problems in the classical LOCAL model.

Quantum distributed algorithms, which allow communication of qubits (with or without pre-shared entanglement), are known to generate output distributions with non-signaling and finite-dependence properties. In particular, prior work established the absence of quantum advantage for 3-coloring unrooted trees via lower bounds in the non-signaling model—the most general model consistent with causality—but left rooted trees open due to technical subtleties. Similarly, for cycles, the best known lower bounds left a gap between classical and quantum round complexity. This paper closes these gaps by demonstrating that quantum resources provide no advantage in distributed 3-coloring of rooted trees or 2-coloring of even cycles.

Results for 3-Coloring Rooted Trees

The main result asserts that any quantum-LOCAL algorithm (without pre-shared entanglement) that properly 3-colors an $n$-node rooted tree with high probability (specifically, at least $1-O(1/\log n)$) requires $\Omega(\log^\star n)$ rounds—exactly matching the classical deterministic lower bounds (Cole-Vishkin and Linial) for rooted trees.

This is achieved by establishing a lower bound of $\Omega(\log^\star \Delta)$ rounds for proper 3-coloring of rooted $\Delta$-ary trees, given the success probability threshold of $1-1/\Delta$, with large $\Delta$. The proof departs from traditional indistinguishability arguments and leverages a color lifting technique inspired by Linial's round-elimination argument, but adapted for the structural properties of quantum-LOCAL output distributions, specifically their finitely-dependent, non-signaling character. The technique exploits the recursive architecture of rooted trees to iteratively lift colorings from lower round complexity, ultimately showing that the output coloring must have a super-constant number of colors unless the protocol runs for $\Omega(\log^\star \Delta)$ rounds.

The results stand even in the presence of adversarial node identifiers and arbitrary node degree, as long as the degree is in the relevant "Ramsey regime" ($\Omega(n)$0 and $\Omega(n)$1). The argument is robust against attempts to generalize quantum lower bounds solely based on non-signaling and finite dependence, as in previous impossibility results for cycles and bounded-degree graphs, thereby resolving a key technical obstacle in the literature.

Results for 2-Coloring Even Cycles

For 2-coloring even-length cycles, the paper proves that any quantum-LOCAL algorithm (even with pre-shared entanglement and unique identifiers) that achieves success probability at least $\Omega(n)$2 requires $\Omega(n)$3 rounds. This sharp lower bound matches the classical deterministic complexity and improves upon earlier non-signaling lower bounds by a factor of two.

The proof circumvents the limitations inherent to non-signaling distributions defined only for cycles by considering a strictly larger family of input graphs (cycles plus cycles with isolated vertices). The analysis establishes that information cannot propagate fast enough (at distance greater than $\Omega(n)$4 in $\Omega(n)$5 rounds) to enable proper coloring, using a reduction from coloring an odd cycle, which always contains an unavoidable monochromatic edge, to the even cycle, thereby forcing the error probability to be at least $\Omega(n)$6 if the round complexity is less than $\Omega(n)$7.

This result tightly precludes any quantum speedup—even by one round—in distributed 2-coloring, regardless of quantum correlations or entanglement, and demonstrates that the quantum-LOCAL model respects the same locality constraints as classical distributed algorithms for even cycles.

Technical Framework

The paper formalizes its results using the quantum-LOCAL computational model, in which nodes can exchange quantum bits and may share pre-entangled states. Output distributions of quantum algorithms are characterized as non-signaling processes, formalizing physical causality—an $\Omega(n)$8-round algorithm’s outputs at any node depend only on their $\Omega(n)$9-neighborhood. Additionally, in the absence of pre-shared global resources, the output processes exhibit finite dependence—outputs at sufficiently distant nodes are independent.

The lower bound proofs rely on the construction of lifted colorings and demonstrate how recursive and structural properties of the problem (in trees and cycles) interplay with output distribution constraints inherent to the quantum-LOCAL model. The paper also provides rigorous mappings from quantum distributed algorithms to non-signaling, finitely-dependent stochastic processes, justifying the universality of the lower bounds in the distributed quantum setting.

Implications and Future Directions

The findings have significant implications for the theory of distributed quantum computing. They categorically rule out quantum advantage for two fundamental locally checkable labeling (LCL) problems, establishing that quantum resources do not overcome the basic locality constraints of the classical LOCAL model for rooted trees and cycles. This settles open questions regarding the round complexity of these problems and delineates the boundary between problems for which quantum speed-up is conceivable and those fundamentally limited by locality (as encoded by Linial's lower bounds and their extensions).

Practically, these results signal that future research in quantum distributed algorithms must focus on problems with genuinely nonlocal structure or those in models (such as CONGEST) where quantum communication enables speed-up via bandwidth bottleneck alleviation, rather than locality relaxation. The paper's round-elimination argument for quantum algorithms suggests pathways to further extend quantum lower bounds for directed cycles and other LCLs by exploiting graph structure and communication limitations in the quantum setting.

Theoretically, the work prompts the re-examination of block-factor and round-elimination techniques in quantum communication complexity, and raises intricate technical questions regarding natural problems versus artificial tasks in quantum distributed algorithm design. It further clarifies the relationships between quantum-LOCAL, randomized online-LOCAL, and non-signaling models, highlighting their respective strengths and limitations.

Conclusion

The paper delivers rigorous, structural proofs that no distributed quantum advantage is possible for 3-coloring rooted trees and 2-coloring even cycles in the quantum-LOCAL model, regardless of quantum resources or pre-shared entanglement. It achieves tight lower bounds, matching the best known classical complexities, and resolves longstanding gaps in the theory. The results advance the understanding of quantum locality and its boundaries in distributed computing, and suggest refined directions for future research in distributed quantum algorithms.