Online Locality Meets Distributed Quantum Computing (2403.01903v3)

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.

• The paper demonstrates that LCL problems solvable in O(log* n) time classically can be addressed with constant locality using finitely-dependent processes.
• The analysis establishes a model hierarchy by showing randomized online-LOCAL methods simulate quantum and classical algorithms efficiently in tree structures.
• The paper provides a lower bound for the three-coloring problem in grids, highlighting the computational limits of advanced distributed models.

Online Locality Meets Distributed Quantum Computing

The paper "Online Locality Meets Distributed Quantum Computing" by Akbari et al. investigates the theoretical underpinnings of locally checkable labeling (LCL) problems within the classical LOCAL model of distributed computing and extends this examination to various state-of-the-art models, including quantum-LOCAL, finitely-dependent processes, and others. The authors provide novel insights into the advantages and limitations of these models, highlighting the implications for distributed quantum computing.

Overview of Models and Extensions

The authors meticulously extend the theory of LCL problems beyond the classical deterministic LOCAL model, which has been extensively studied and characterized. Central to their analysis is the quantum-LOCAL model, which allows node processors to utilize quantum information and quantum communication channels. This model, positioned as a superset of the classical randomized LOCAL model, leverages quantum mechanics to potentially offer computational advantages.

The non-signaling model is another important focus. It leverages non-signaling principles from theoretical physics, ensuring that distant parts of a distributed system cannot affect each other's states instantaneously, thereby respecting causality. Similarly, the authors address finitely-dependent processes, which are stochastic processes where dependencies decay beyond a certain distance—an essential concept for understanding symmetry-breaking LCLs.

Moreover, the paper explores the dynamic-LOCAL and online-LOCAL models, both vital for studying algorithm performance under adversarial conditions. These models simulate realistic dynamic updates and online processing of graph nodes, reflecting real-world scenarios where the input graph can change or be revealed incrementally.

Fundamental Contributions

The paper’s contributions can be distilled into three primary results:

1. Demonstrating the Superiority of Finitely-Dependent Processes: The authors show that any LCL problem solvable within the classical deterministic LOCAL model with locality $O(\log^* n)$ can also be addressed using finitely-dependent processes with constant locality. This is a significant finding as it bridges the gap between classical and probabilistic methods, illustrating that classical symmetry-breaking constraints can be managed efficiently using probabilistic techniques. This directly advances the understanding of distributed quantum advantage limitations by showing these classical problems can be effectively resolved without relying on non-signaling arguments.
2. Hierarchical Comparison and New Boundaries: The hierarchical classification of models provides a structured comparison of their capabilities. Specifically, they introduce a randomized online-LOCAL model capable of simulating quantum and classical algorithms effectively. It demonstrates that in trees, solving LCL problems with locality $o(n^{(5)})$ in the randomized online-LOCAL model implies a solution in the deterministic LOCAL model within $O(\log^* n)$ locality. This result highlights the robustness and computational power of the online-LOCAL model in constrained graph settings, underlining that certain hard deterministic problems retain their complexity even under more liberal models like quantum-LOCAL.
3. Three-Coloring in Grids and Lower Bounds: By proving a lower bound for the three-coloring problem in bipartite grids within the randomized online-LOCAL model, the authors provide strong evidence of the computational limits in such advanced models. This not only affirms that randomness does not provide additional leverage in certain distributive graph problems but also sets a precedent for future hardness proofs in the dynamic behaviors of complex graph structures.

Practical and Theoretical Implications

The results presented in this paper have profound implications for both theoretical research and practical algorithm design. From a theoretical perspective, the exploration of LCL problems across varied computational models deepens the understanding of distributed computing, particularly in mitigating quantum advantages in distributed settings. Practically, these findings inform the development of more efficient algorithms under realistic constraints where the input is not wholly available initially, banishing the notion of absolute superiority of quantum models for specific categories of problems.

Conclusion and Future Directions

Akbari et al.'s paper provides a comprehensive examination of LCL problems across a spectrum of computational models, revealing intriguing connections and new frontiers for both classical and quantum distributed computing. The implications of finitely-dependent processes and the robust nature of the online-LOCAL model stand out as noteworthy contributions. Future research should leverage these insights to explore LCL problems in more complex graph structures and further investigate the boundaries of quantum computational advantages within distributed frameworks. This is particularly relevant as real-world applications increasingly demand distributed solutions that adapt dynamically and scale efficiently with quantum computing capabilities.