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Density Matrix-based Dynamics for Quantum Robotic Swarms

Maria Mannone, Mahathi Anand, Peppino Fazio, Abdalla Swikir

Abstract

In a robotic swarm, parameters such as position and proximity to the target can be described in terms of probability amplitudes. This idea led to recent studies on a quantum approach to the definition of the swarm, including a block-matrix representation. However, the size of such matrix-based representation increases drastically with the swarm size, making them impractical for large swarms. Hence, in this work, we propose a new approach for modeling robotic swarms and robotic networks by considering them as mixed quantum states that can be represented mathematically via density matrices. The size of such an approach only depends on the available degrees of freedom of the robot, and not its swarm size and thus scales well to large swarms. Moreover, it also enables the extraction of local information of the robots from the global swarm information contained in the density matrices, facilitating decentralized behavior that aligns with the collective swarm behavior. Our approach is validated on several simulations including large-scale swarms of up to 1000 robots. Finally, we provide some directions for future research that could potentially widen the impact of our approach.

Density Matrix-based Dynamics for Quantum Robotic Swarms

Abstract

In a robotic swarm, parameters such as position and proximity to the target can be described in terms of probability amplitudes. This idea led to recent studies on a quantum approach to the definition of the swarm, including a block-matrix representation. However, the size of such matrix-based representation increases drastically with the swarm size, making them impractical for large swarms. Hence, in this work, we propose a new approach for modeling robotic swarms and robotic networks by considering them as mixed quantum states that can be represented mathematically via density matrices. The size of such an approach only depends on the available degrees of freedom of the robot, and not its swarm size and thus scales well to large swarms. Moreover, it also enables the extraction of local information of the robots from the global swarm information contained in the density matrices, facilitating decentralized behavior that aligns with the collective swarm behavior. Our approach is validated on several simulations including large-scale swarms of up to 1000 robots. Finally, we provide some directions for future research that could potentially widen the impact of our approach.

Paper Structure

This paper contains 14 sections, 61 equations, 7 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Notations
  4. Preliminaries
  5. Quantum Robotic Swarm Modeling
  6. Density Matrix-based Modeling of Quantum Robotic Swarm
  7. Examples
  8. Pure States
  9. Mixed States
  10. Other Computational Examples
  11. Movements on a 2D surface, First Examples with Qiskit, and Larger Swarms
  12. Time Evolution
  13. Distributed Quantum Swarm
  14. Conclusions and Discussions

Figures (7)

  • Figure 1: Pictorial representation of the transition from a classic robotic swarm to a quantum-robotic swarm: in a given arena, each robot is considered in terms of a probability amplitude to be found in a certain space position.
  • Figure 2: Quantum-like representation of a toy 2-robot swarm, where the position of the robot is considered as the weighted peak of the corresponding wavefunction.
  • Figure 3: Wavefunction visualization of the probability amplitudes to find the two robots-particles described in Eq. \ref{['density_matrix_initial_state_working']} and \ref{['density_matrix_final_state_working']}, respectively.
  • Figure 4: Visualization of the 10-robot swarm at the beginning of our simulation. The turquoise star indicates the target, adopting the style of 2D representation of swarm_paper.
  • Figure 5: Visualization of the 10-robot swarm once it reached the target.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Definition 1
  • Remark 1
  • Remark 2