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Quantum Strategies for Rendezvous and Domination Tasks on Graphs with Mobile Agents

Giuseppe Viola, Piotr Mironowicz

TL;DR

This work demonstrates that quantum non-locality can enhance distributed coordination among mobile agents on graphs, by introducing rendezvous and a domination-inspired task and casting them as Bell games. It employs SDP-based tools (NPA hierarchy) and the see-saw method to quantify and realize quantum strategies, comparing them against classical limits. Across a variety of graphs, two-agent quantum strategies show clear advantages in rendezvous and, in some cases, domination, while three-agent scenarios often show no observed benefit, highlighting a nuanced, graph-dependent landscape. The results have implications for quantum-enabled coordination in networks and robotics, and point to rich avenues for future analytical and computational exploration of quantum advantages in distributed tasks.

Abstract

This paper explores the application of quantum non-locality, a renowned and unique phenomenon acknowledged as a valuable resource. Focusing on a novel application, we demonstrate its quantum advantage for mobile agents engaged in specific distributed tasks without communication. The research addresses the significant challenge of rendezvous on graphs and introduces a new distributed task for mobile agents grounded in the graph domination problem. Through an investigation across various graph scenarios, we showcase the quantum advantage. Additionally, we scrutinize deterministic strategies, highlighting their comparatively lower efficiency compared to quantum strategies. The paper concludes with a numerical analysis, providing further insights into our findings.

Quantum Strategies for Rendezvous and Domination Tasks on Graphs with Mobile Agents

TL;DR

This work demonstrates that quantum non-locality can enhance distributed coordination among mobile agents on graphs, by introducing rendezvous and a domination-inspired task and casting them as Bell games. It employs SDP-based tools (NPA hierarchy) and the see-saw method to quantify and realize quantum strategies, comparing them against classical limits. Across a variety of graphs, two-agent quantum strategies show clear advantages in rendezvous and, in some cases, domination, while three-agent scenarios often show no observed benefit, highlighting a nuanced, graph-dependent landscape. The results have implications for quantum-enabled coordination in networks and robotics, and point to rich avenues for future analytical and computational exploration of quantum advantages in distributed tasks.

Abstract

This paper explores the application of quantum non-locality, a renowned and unique phenomenon acknowledged as a valuable resource. Focusing on a novel application, we demonstrate its quantum advantage for mobile agents engaged in specific distributed tasks without communication. The research addresses the significant challenge of rendezvous on graphs and introduces a new distributed task for mobile agents grounded in the graph domination problem. Through an investigation across various graph scenarios, we showcase the quantum advantage. Additionally, we scrutinize deterministic strategies, highlighting their comparatively lower efficiency compared to quantum strategies. The paper concludes with a numerical analysis, providing further insights into our findings.
Paper Structure (23 sections, 4 theorems, 27 equations, 5 figures, 5 tables)

This paper contains 23 sections, 4 theorems, 27 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Distributed Tasks for Agents
  3. Rendezvous on Graphs
  4. Domination on Graphs
  5. Considered Scenarios and Definitions
  6. Classical and Quantum Probabilities and Bell Games
  7. Distributed Tasks as Bell Games
  8. Numerical Optimization Methods
  9. Semi-definite Programming
  10. The Navascués-Pironio-Acı́n Method
  11. The See-Saw Method
  12. Properties of Symmetric Deterministic Strategies for Distributed Tasks
  13. Symmetric Deterministic Strategies for Rendezvous on Graphs
  14. Deterministic strategies for Domination on Graphs
  15. Results
  16. ...and 8 more sections

Key Result

Lemma 1

Symmetric strategies for the rendezvous task are at least as good as non-symmetric strategies.

Figures (5)

  • Figure 1: Some of the graphs analysed in this work.
  • Figure 2: Graph with a 3-line curly structure.
  • Figure 3: Graph with the "hat" structure.
  • Figure 4: (color online) Pentagon with optimal classical strategies of Alice and Bob for domination. Red arrows denote moves of one of the parties, and blue arrows moves of the other party.
  • Figure 5: (color online) Pentagon with optimal quantum strategies of Alice and Bob for domination. Arrows denote the moves of the parties in different cases, and the labels refer to probabilities of the movement; see the text in sec. \ref{['ssec:pentagon']} for the description.

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • Lemma 3
  • proof