# Categories, Quantum Computing, and Swarm Robotics: A Case Study

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## Abstract

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## 1. Introduction

#### 1.1. Swarms of Robots

#### 1.2. Ideal Swarms

- Can work in whatever scenario (e.g., on the ground, underwater, on-air);
- Can adapt to any possible adverse condition (during a storm, after an earthquake, a tsunami, an avalanche);
- Is able to communicate in any imaginable way (though laser, infrared light, visible light, radar, sonar);
- Is ready for whichever task (e.g., search and rescue, object retrieval, stacking together to perform a more complex task).

## 2. Theoretical Framework and Methods

- Signals transmitted from ${R}_{1}$ to ${R}_{2}$ and reactions of ${R}_{2}$ (term ${R}_{1}\ast {R}_{2}$),
- Signals transmitted from ${R}_{2}$ to ${R}_{1}$ and reactions of ${R}_{1}$ (term ${R}_{2}\ast {R}_{1}$), respectively.

OPENQASM 2.0; include “qelib1.inc” qreg q[3]; creg c[1]; //classic qubit for the measure //states are initialized as 0 by default. ry(pi/2) q[0]; // to obtain 0.5, 0.5 as amplitudes //x q[1]; // to obtain input amplitude 0.0, 1.0, that is, eigenstate |1> ry(1.2309594) q[1]; // for input amplitudes 0.7, 0.3 // 1.9106332 for input amplitudes 0.3, 0.7 barrier q[0], q[1], q[2]; ccx q[0], q[1], q[2]; x q[0]; x q[1]; ccx q[0], q[1],q[2]; x q[0]; x q[1]; barrier q[0], q[1], q[2]; measure q[2] -> c[0];

## 3. Results

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The considered quantum circuit for $|{q}_{0}\rangle =\frac{1}{\sqrt{2}}|0\rangle +\frac{1}{\sqrt{2}}|1\rangle ,\phantom{\rule{0.166667em}{0ex}}|{q}_{1}\rangle =\sqrt{\frac{2}{3}}|0\rangle +\frac{1}{\sqrt{3}}|1\rangle $. Changing the initializations of ${q}_{0}$ and ${q}_{1}$ through suitable ${R}_{y}$ gates, we can span the different cases.

**Table 1.**Truth tables (reversible equivalents of XNOR gates), representing the interaction between robot 1, ${R}_{1}$ (${q}_{0}$: position, ${q}_{1}$: reward) and robot 2, ${R}_{2}$ (${q}_{2}$: position, ${q}_{3}$: reward). At the beginning (left table), ${R}_{1}$ communicates its position (down/up) and reward (no/yes); then it waits, and, according to its information, ${R}_{2}$ reaches ${R}_{1}$ or not (when ${R}_{2}$ reaches ${R}_{1}$, ${q}_{2}$ becomes equal to ${q}_{0}$). Then (right table), we have the symmetric situation, where ${R}_{2}$ communicates position and reward, and ${R}_{1}$ decides to reach it nor not (if ${R}_{1}$ reaches ${R}_{2}$, ${q}_{0}$ becomes equal to ${q}_{2}$). Equation (8) shows the corresponding permutation matrix.

Gate ${\mathit{t}}_{0}\to {\mathit{t}}_{1}$ | Gate ${\mathit{t}}_{1}\to {\mathit{t}}_{2}$ | ||||||
---|---|---|---|---|---|---|---|

input | output | input | output | ||||

${\mathit{q}}_{\mathbf{0}}$ | ${\mathit{q}}_{\mathbf{1}}$ | ${\mathit{q}}_{\mathbf{0}}$ | ${\mathit{q}}_{\mathbf{2}}$ | ${\mathit{q}}_{\mathbf{2}}$ | ${\mathit{q}}_{\mathbf{3}}$ | ${\mathit{q}}_{\mathbf{2}}$ | ${\mathit{q}}_{\mathbf{0}}$ |

0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |

0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 |

1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

**Table 2.**Comparisons of measurement outcomes for 1024 shots of the code. S indicates the simulator; C stands for a real quantum computer located in Bogotà, Colombia.

$|{\mathit{q}}_{0}\rangle $ | $|{\mathit{q}}_{1}\rangle $ | $|{\mathit{q}}_{2}\rangle $ (Expected) | Counts for $0,\phantom{\rule{0.166667em}{0ex}}1$ | Device |
---|---|---|---|---|

$|1\rangle $ | $|1\rangle $ | 1 | 0; 1024 | S |

455; 569 | C | |||

$|1\rangle $ | $|0\rangle $ | 0 | 1024; 0 | S |

527; 497 | C | |||

$\frac{1}{\sqrt{2}}|0\rangle +\frac{1}{\sqrt{2}}|1\rangle $ | $|1\rangle $ | uncertain | 494; 530 | S |

502; 522 | C | |||

$\frac{1}{\sqrt{2}}|0\rangle +\frac{1}{\sqrt{2}}|1\rangle $ | $|0\rangle $ | uncertain | 527; 497 | S |

431; 593 | C | |||

$|0\rangle $ | $\sqrt{\frac{2}{3}}|0\rangle +\frac{1}{\sqrt{3}}|1\rangle $ | 0 | 686; 338 | S |

531; 493 | C | |||

$|1\rangle $ | $\sqrt{\frac{2}{3}}|0\rangle +\frac{1}{\sqrt{3}}|1\rangle $ | 0 | 702; 322 | S |

570; 414 | C | |||

$|0\rangle $ | $\frac{1}{\sqrt{3}}|0\rangle +\sqrt{\frac{2}{3}}|1\rangle $ | 1 | 315; 709 | S |

452; 572 | C | |||

$|1\rangle $ | $\frac{1}{\sqrt{3}}|0\rangle +\sqrt{\frac{2}{3}}|1\rangle $ | 1 | 335; 669 | S |

376; 648 | C | |||

$\frac{1}{\sqrt{2}}|0\rangle +\frac{1}{\sqrt{2}}|1\rangle $ | $\sqrt{\frac{2}{3}}|0\rangle +\frac{1}{\sqrt{3}}|1\rangle $ | uncertain | 518; 506 | S |

351; 673 | C | |||

$\frac{1}{\sqrt{2}}|0\rangle +\frac{1}{\sqrt{2}}|1\rangle $ | $\frac{1}{\sqrt{3}}|0\rangle +\sqrt{\frac{2}{3}}|1\rangle $ | uncertain | 508; 516 | S |

433; 591 | C |

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Mannone, M.; Seidita, V.; Chella, A.
Categories, Quantum Computing, and Swarm Robotics: A Case Study. *Mathematics* **2022**, *10*, 372.
https://doi.org/10.3390/math10030372

**AMA Style**

Mannone M, Seidita V, Chella A.
Categories, Quantum Computing, and Swarm Robotics: A Case Study. *Mathematics*. 2022; 10(3):372.
https://doi.org/10.3390/math10030372

**Chicago/Turabian Style**

Mannone, Maria, Valeria Seidita, and Antonio Chella.
2022. "Categories, Quantum Computing, and Swarm Robotics: A Case Study" *Mathematics* 10, no. 3: 372.
https://doi.org/10.3390/math10030372