No distributed quantum advantage for approximate graph coloring (2307.09444v3)

Abstract: We give an almost complete characterization of the hardness of $c$-coloring $\chi$-chromatic graphs with distributed algorithms, for a wide range of models of distributed computing. In particular, we show that these problems do not admit any distributed quantum advantage. To do that: 1) We give a new distributed algorithm that finds a $c$-coloring in $\chi$-chromatic graphs in $\tilde{\mathcal{O}}(n^{\frac{1}{\alpha}})$ rounds, with $\alpha = \bigl\lfloor\frac{c-1}{\chi - 1}\bigr\rfloor$. 2) We prove that any distributed algorithm for this problem requires $\Omega(n^{\frac{1}{\alpha}})$ rounds. Our upper bound holds in the classical, deterministic LOCAL model, while the near-matching lower bound holds in the non-signaling model. This model, introduced by Arfaoui and Fraigniaud in 2014, captures all models of distributed graph algorithms that obey physical causality; this includes not only classical deterministic LOCAL and randomized LOCAL but also quantum-LOCAL, even with a pre-shared quantum state. We also show that similar arguments can be used to prove that, e.g., 3-coloring 2-dimensional grids or $c$-coloring trees remain hard problems even for the non-signaling model, and in particular do not admit any quantum advantage. Our lower-bound arguments are purely graph-theoretic at heart; no background on quantum information theory is needed to establish the proofs.

• The paper introduces an advanced algorithm that computes c-coloring in χ-chromatic graphs within Õ(n^(1/α)) rounds in the deterministic LOCAL model.
• It establishes a lower bound of Ω(n^(1/α)) rounds for any distributed algorithm, confirming optimality up to polylogarithmic factors across models including quantum.
• The work conclusively demonstrates no quantum advantage in distributed graph coloring, reshaping expectations for quantum improvements in similar tasks.

No Distributed Quantum Advantage for Approximate Graph Coloring

The paper addresses the complexity of distributed algorithms for $c$-coloring $\chi$-chromatic graphs, examining the implications across various models of distributed computing, including deterministic, randomized, and quantum models. The central assertion is that distributed quantum computing does not offer an advantage over classical methods in this context, a conclusion supported by a combination of new algorithmic strategies and graph-theoretic arguments.

Key Contributions

1. Algorithmic Upper Bound: The research introduces an advanced algorithm that computes a $c$-coloring in $\chi$-chromatic graphs within $\tilde{\mathcal{O}}(n^{1/\alpha})$ rounds, where $\alpha = \floor*{(c-1)/(\chi-1)}$. This result holds in the deterministic LOCAL model, setting a high-water mark for what can be achieved using classical distributed techniques.
2. Lower Bound Analysis: It is established that any distributed algorithm for the given problem requires at least $\Omega(n^{1/\alpha})$ rounds. This lower bound is crucial as it matches the upper bound, thus demonstrating optimality up to polylogarithmic factors. The results are applicable across several models, including the non-signaling model, which captures the quantum-LOCAL paradigm.
3. No Quantum Advantage: The paper decisively shows that there is no significant advantage in using quantum resources for these coloring tasks. This conclusion expands the understanding of quantum computing's role in distributed algorithms, aligning with conjectures that quantum doesn’t always outperform classical methods for such problems.

Graph-Theoretic Foundations

The research employs sophisticated graph-theoretic constructs to both prove the lower bounds and demonstrate the complexity of approximate coloring. This involves the use of high-girth and high-chromatic graphs, particularly the constructs from Bogdanov's work, as well as quadrangulations of surfaces like the Klein bottle. These constructions showcase the difference between local and global graph properties, leading to high complexity when attempting distributed solutions.

Implications and Future Directions

The implications of these findings are profound, suggesting that pursuit of quantum advantage in distributed settings may require new directions or problems. This work also invites further exploration into the boundary between problems where quantum computing offers benefits and those where it does not.

The theoretical contribution aligns closely with ongoing research in distributed computing and quantum algorithms. Future work could explore other NP-hard graph problems to determine if similar conclusions hold, or investigate how relaxing certain problem constraints might change the scenario.

In summary, the paper provides a deep dive into the complexities of distributed graph coloring, reinforcing that quantum algorithms, while promising for some domains, do not universally surpass classical strategies in distributed computational settings.