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Quantum RoboSound:

Auditory Feedback of a Quantum-Driven Robotic Swarm

Maria Mannone1,2, Valeria Seidita1, and Antonio Chella1,3

Abstract— Data soniﬁcation enhance and enrich information

understanding with an additional sensory dimension. Soniﬁca-

tion also opens the way to more creative applications, joining

arts and sciences. In this study, we present sequences of chords

obtained as auditory feedback from the trajectories of a robotic

swarm. The swarm behavior is an emerging effect from simple

local interactions and autonomous decisions of each robot. The

swarm effect can be identiﬁed through soniﬁcation outcomes in

terms of voice leading patterns. Thus, chord patterns represent

behavior patterns. The convergence to the target is represented

by the convergence to a speciﬁc pitch. The swarm decision

process is based upon quantum computing. We ﬁrst present

logic gates and their implementations as quantum circuits,

describing examples with 2- and 3-dimensional motion of a

3-robot toy swarm. The considered scenarios are ant foraging

(2D) and underwater search and rescue (3D). Then, we provide

and discuss some examples of the harmonic sequences that can

be obtained as feedback from robotic motion.

I. INTRODUCTION

The whole is more than the sum of the single parts: that

could be the beginning of a chapter on Gestalt, as well as of

a discussion on swarms. Be they ﬂocking birds [1], schooling

ﬁsh [2], termites building a complex nest [3], bacteria

silently working inside the human body, self-organizing cells

painting the stripes of a tiger [4], natural swarms fascinate

and inspire. A natural swarm is a self-organized collection

of agents (animals, bacteria, cells), where interacting single,

simple contributions give rise to a complex phenomenon.

We can thus talk of swarm intelligence [5] as an instance of

distributed intelligence.

From nature to human artifacts, there are numerous at-

tempts to imitate nature through robotic swarms [6], [7].

Natural swarms inspired robotic ﬁsh coordination [8], micro-

swarms used for therapeutical reasons [9], tools to investigate

pattern-making in morphogenesis [4]. A robotic swarm is a

self-organized [10], scalable [6], decentralized [11] system

made up of single interacting entities. Even though individual

*The research leading to these results takes place within the framework

of the project “ARES, Autonomous Robotics for the Extended Ship,” funded

by the Italian Ministry of University and Research under grant agreement

ARS01 00682.

1Valeria Seidita is assistant Professor at the Department of Engineering

of the University of Palermo, Italy valeria.seidita@unipa.it

2Maria Mannone is postdoctoral researcher at the Department

of Engineering of the University of Palermo, Italy, and subject

expert at DAIS and ECLT of Ca’ Foscari University of

Venice, Italy mariacaterina.mannone@unipa.it,

maria.mannone@unive.it

2Antonio Chella is full Professor at the Department of Engineering of the

University of Palermo, Italy, and professor at ICAR-CNR National Research

Council, Palermo, Italy antonio.chella@unipa.it

tasks are simple, as the result of collaboration, complex goals

can be achieved.

Along with scientiﬁc developments, swarms also inspire

artistic and aesthetic applications [12]. In particular, music

can provide a feedback of robotic swarm and, more in

general, multi-robot behavior [13]. In music, the orchestra

can be seen as an instance of interacting agents (musicians)

with a leader (conductor). Thus, it can be modeled through a

centralized and heterogeneous swarm. In a recent study [14],

robots in a swarm aggregate according to their emitted timbre

similarity. In our research, robots autonomously choose their

positions, that are then mapped to sound. Position proximity

corresponds to pitch proximity. In this study, we are driven

by studies on voice leading [15], that, in the robotic context,

can help identify patterns of behavior.

Nature counts different kinds of swarms; the same happens

in human-made and nature-inspired swarms. To compare

different robotic swarms between them, we can use a formal-

ism derived from category theory [16], an abstract branch

of mathematics [17], [18]. Focusing on a speciﬁc robotic

swarm, we can connect the global swarm behavior with

the local, individual behavior through block matrices [16].

In these nested matrices, diagonal sub-matrices correspond

to each single robot (sensor activation, sending-receiving

functions, position and reward self-information), and off-

diagonal sub-matrices are the pairwise interaction terms.

Aiming to give them a speciﬁc form, we can use the idea

of truth table, with inputs (position, reward) and output

(suggested new position). Implementing truth tables through

reversible logic gates, we can exploit the computational

resources of quantum computing.

Quantum computing [19], [20] is a branch of computer sci-

ence based on the the basic principles of quantum mechanics,

which can be simulated on classical devices or implemented

on real quantum computers. Quantum computing is more and

more applied to artiﬁcial intelligence [21].

The application of computing to robots has been ﬁrst the-

oretically proposed in physics [22], [23], and it is nowadays

implemented to develop new and fast algorithms [24], [25],

enhance communication strategies via entangled states [26],

[27], improve swarm optimization algorithms [28].

More recently, there have been pioneering applications of

quantum computing to robotic swarms [29], [16]. Quantum

logic can be seen as a particular case of fuzzy logic [30],

which has also been applied to robotic swarms [31] for an

underwater scenario. Future developments of robotic swarms

[32], jointly with quantum computing, can lead to quantum

reinforcement learning [33], [34]. In [16], a new quantum

circuit has been envisaged and implemented for a 2-robot

toy swarm moving along a line. This model has then been

uplifted to two dimensions, with the robot moving on a

plane [35], as the ants moving back and forth between the

nest and the food source [36]. Ants already inspired robotic

applications [37] and, more in general, hormone-inspired

robotic versions of self-organized systems [38].

In this research, we propose the extension of the circuit to

three dimensions, modeling a motion in the 3D space, and

a soniﬁcation of 2D and 3D models to obtain an auditory

feedback for these processes. We used IBM QASM simulator

for our examples. We use quantum computing to build a

decisional system that, thanks to quantum efﬁciency, may

render collective decisions of robotics swarms faster. We use

soniﬁcation to obtain a more intuitive grasping of the overall

swarm movement. The combination of the two can thus

produce a faster swarm decision and more clear perception

of its effect as collective motion.

The article is organized as follows. In Section II, we

introduce the basics of the exploited quantum formalism

and decision process for 2 dimensions, with a logic gate

and the corresponding circuit. In Section III, we provide the

circuit and the logic gate needed for 3-dimensional motion.

The soniﬁcation strategy is presented in Section IV, jointly

with the link to access sound samples. Discussions and

Conclusions (Section V) end the article.

II. BACKGROU ND

In this Section, we ﬁrst brieﬂy present the basic concepts

of quantum computing we exploit in our study. Then, we

summarize an application of quantum computing to model a

robotic swarm decision-making for 2-dimensional motion.

We use the quantum idea of state superposition to model

the positions along the axes and the reward. Drawing upon

the idea of wavefunction with states’ superposition, in eq.

(1), we consider the position along xas a superposition of

state |0i, left, and |1i, right, with probability amplitudes αx

and βx, respectively.

|q0i=αx|0i+βx|1i(1)

The same reasoning is to be adopted for the positions along

yand z-axis. Similarly, we deﬁne the reward, evaluated here

as a measure of proximity to the target, as the quantum

superposition of |1i, success, and |0i, failure, see eq. (2).

|q1i=γ|0i+δ|1i(2)

In 2- and 3-dimensional examples, we consider a motion on

the surface x∈[0,1], y ∈[0,1] and space with z∈[0,1].

We compute the reward as 1−the Euclidean distance from

the target: the closer the target, the closer the reward to 1; the

farther the target, the closer the reward to 0. In our models,

we consider the search and rescue swarm task.

The other basic concept we consider is the quantum

measure [20]. Quantum measure is destructive, because it

provokes the collapse of the wavefunction to a particular state

[19]. The circuits we present in the following contain, at the

end, the measure operation, which is performed through a

quantum simulator or a real quantum computer. There are

1024 shots for each circuit, and the more the occurrences of

a particular outcome, the more likely that state superposition.

For 2D and 3D models, we included a ﬁnal step of

entanglement. The quantum phenomenon of entanglement

[39], [40] is, in quantum computing, an invaluable resource

to speed up computations [41]. The use of entanglement has

been proposed to connect individual behaviors in a robotic

swarm [23]; we can use it as a later step of connection

[35]. In particular, we focus on the GHZ state [42], to help

all robotic positions switch in a synchronized way between

a position and another one, according to the result of the

measure.

The presented circuits for 2D and 3D cases have been

embedded in a Python-based Jupyter Notebook, with

further steps of positions reshufﬂe (when all robots have

a too-low reward) and movement toward the robot with

the highest reward, to reﬁne the procedure. The QISKIT

code for 1D and the QISKIT and Jupyter codes for 2D

can be found in our GitHub repositories https://

github.com/medusamedusa/quantum-world and

https://github.com/medusamedusa/3-robot,

respectively.

The 1-dimensional case, with its logic gate and the associ-

ated quantum circuit, the reversible equivalent of an XNOR

gate, is presented in [16]. The core idea is: if the reward of

Riis high, reach it; otherwise, go explore somewhere else.

Let us focus on a 2-dimensional motion of a 3-robot toy

swarm, presenting the truth table (Table I) and the circuit

implementing it (Figure 1). Table I contains the following

elements:

•|q0iis the x-position of Riat t1;

•|q1iis the y-position of Riat t1;

•|q2iis the reward of Riat t1;

•|q3iis the suggested x-position of Rjat t2;

•|q4iis the suggested y-position of Rjat t2.

Thus, the inputs are xy-position and reward of Ri, and the

outputs include the suggested xy-position of Rj, but not

its reward. It is calculated once Rjeffectively changes its

localization reaching the suggested new position. The gate

also returns the reward of Riby just copying it. In this way, if

we restrict to cases with reward =|1i, the gate is reversible.

TABLE I

TRUT H TAB LES FO R TWO ROB OTS Ri, RjON T HE xy-PL AN E

q0q1q2q4q3q2

x-pos y-pos reward y-pos x-pos reward

RiRiRiRjRjRi

0 0 0 0/1 0/10

0 0 1 0 0 1

0 1 0 0/1 0/1 0

0 1 1 1 0 1

1 1 1 1 1 1

1 0 0 0/1 0/1 0

1 1 0 0/1 0/1 0

1 0 1 0 1 1

Fig. 1. Circuit implementing the non-reversible logic gate of truth table

in Table I. The image also presents the example of a successful robot on 1

along y.

However, reversibility is not always veriﬁed. In fact, while

the logic gate proposed for the 1-dimensional case was

reversible, the 2D truth table in its wholeness is no longer

reversible because of the indeterminacy on x, y in the case of

0reward. The outcome indeterminacy, with a superposition

of 0 and 1 as outputs, is in fact in more than one coordinate.

Such an indeterminacy can be solved by considering infor-

mation from the other robots: if another robot got “success”

in some different region of the plane, it can be followed by

the other robots. Thus, if a robot got 0as reward, another

one could get 1and be a better candidate to enter the gate.

In general, the convenient strategy is constituted by cycles

of: exploration, broadcast communication of the exploration

outcome, reaching the most successful robot, reshufﬂe if no

one is successful. We implement this strategy in our Jupyter

notebook codes. Robotic trajectories can be found in the

shared repositories on GitHub.

We are considering state superpositions, so in general we

have a probability amplitude to get failure or success, see eq.

(2). The robot with the highest probability of success is the

selected one to enter the gate. Anyway, within the quantum

circuit implementing the 2D gate (Figure 1), there are single

reversible gates, for deﬁnition of quantum circuit.

In the case of three robots, according to the matrix scheme

in [16], there are six off-diagonal sub-matrices, correspond-

ing to six (3!) interaction terms. However, it is sufﬁcient to

just enable the computation of one term rather than six. In

fact, as said above, the robot entering the circuit is the one

with the highest reward. The more likely output of the circuit

is the most probable position for all the other robots at the

following time point. This case is described in [35].

For our 2D implementation, we chose a biological-inspired

ant foraging scenario [36], with ants moving back and forth

between the nest and the food source.

III. A NE W MO D EL : 3-DIMENSIONAL MOTION

Let us keep focusing on a 3-robot swarm, this time

moving throughout the 3-dimensional space. We still have

six interaction terms. The truth table for the space motion

is given in Table II, and it is implemented by the circuit in

Figure 2. Table II contains the following elements:

•|q0iis the x-position of Riat t0;

•|q1iis the y-position of Riat t0;

•|q2iis the z-position of Riat t0;

•|q3iis the reward of Riat t0;

•|q4iis the suggested x-position of Rjat t1;

•|q5iis the suggested y-position of Rjat t1;

•|q6iis the suggested z-position of Rjat t1.

TABLE II

TRUT H TAB LE S FO R TW O ROB OTS Ri, RjO N TH E xyz -S PACE

q0q1q2q3q6q5q4q3

x-pos y-pos z-pos reward z-pos y-pos x-pos reward

RiRiRiRiRjRjRjRi

1 1 1 1 1 1 1 1

0 1 1 1 1 1 0 1

0 0 0 1 0 0 0 1

0 1 0 1 0 1 0 1

1 0 1 1 1 0 1 1

1 0 0 1 0 0 1 1

1 1 0 1 0 1 1 1

0 0 0 1 0 0 0 1

0 0 1 1 1 0 0 1

1 0 0 1 0 0 1 1

1 1 1 0 0/1 0/1 0/1 0

0 1 1 0 0/1 0/1 0/1 0

... ... ... ... ... ... ... ...

Fig. 2. Circuit implementing the non-reversible logic gate of truth table

in Table II.

The code with the updated Jupyter notebook and

the circuit simulations screenshots can be found at

https://github.com/medusamedusa/quantum_

robo_sound

The scenario we consider here is an underwater search and

rescue task. We examine the case of aquatic robots starting

from a ship and reaching the target in the ocean abyss (video

http://tiny.cc/9ctouz).

IV. SCHEME OF SONIFICATION AND RESULTS

Let us now describe the soniﬁcation strategy adopted to

convert robots’ positions into sound, and analyze displace-

ments as voice leading variations. Each robot is associated

with a speciﬁc instrument: thus, simple timbre variations

allow the identiﬁcation of a speciﬁc robot in the overall

resulting chord. At each time point, every robot sends a

musical note, corresponding to the position. Thus, for a 3-

robot swarm, we have a 3-note chord (a triad) at each time.

Auditory outcome in terms of chords is an emergent effect,

providing information on the swarm behavior. For example,

the parallel motion of robots has its equivalent in a parallel

motion of voices across chords. The convergence of robots

toward the same spatial point corresponds to a convergence

toward the same pitch. We can have the case where robots

are showing an organized behavior moving together (in a

parallel way or converging toward a point), but the ending

point/region of space is far from the target. We can also

have the case of a target reached through a non-coordinated

motion—this case is avoided in our modeling, because of

the ﬁnal step of entanglement. In music, the theory of voice

leading [15] can allow us analyze auditory outcomes and

identify patterns of behavior.

A. Soniﬁcation of 2D-motion

To create our 2D soniﬁcations, we approximately mapped

the 12-note circle to the axes. The circle contains 12 evenly-

distributed points (as in a clock), the notes of a chromatic

scale as pitch classes (modulo 12), see Figure 3 (a). To

prepare musical examples, we chose speciﬁc discrete pitches.

We made the note D coincide with the starting point for

robots (the nest position in the ant example), and F], its

antipodal point, as the target (the food-source position). A

more ﬁne-graining soniﬁcation requires continuous pitches.

We associate a different instrument to each robot, to keep

auditory distinction between individual paths. We chose

trumpet, ﬂute, and cello (with cello an octave lower acting

as the bass).

A

F#

G

A#

B

C

D#

E

F

C#

D

x

y

G#

above 0.5

below 0.5

0

1

0.17 1

0.3 0.5

0.5

0.67 0.84

(a) (b)

above 0.5

below 0.5

0

1

0.5

z

x

y

(c)

Fig. 3. The considered pitch scheme.

Transcribing our chord outputs in musical staff, we neglect

repetitions of consecutive identical chords. In fact, we added

a sound output section after each block of code. In the case

of not called-out if-based instructions, the robots’ positions

are unchanged, and the chord remains the same. Classic pro-

hibition of hidden or manifest parallel ﬁfths/octaves cannot

of course be considered here. We obtain a dissonant 3-part

harmony. With 4 robots, we could obtain a 4-part harmony.

Our 2D examples present a real-sound notation (for trumpet,

the only transposing instrument in our 2D samples, we write

the pitch that is actually played).

In the ﬁrst example, we show an example of an off-target

convergence (reward 0.7; we consider success if δ≥0.8);

see the score in Figure 4. From the voice leading, it is evident

that the parallel motion of R1and R3in bar 1, the parallel

motion of R2and R3between the second chord of the ﬁrst

bar and the ﬁrst chord of the second bar, and the parallel,

descending motion of these two robots toward the ﬁnal chord.

In the last bar, we have a unison on F, one tone away from the

correct target at F]. Interestingly, the sequence gives a feeling

of musical cadence, as some of the next examples, because of

the progressive approaching of the robots and the required

step of GHZ entanglement. In these examples, dissonance

and consonance (with the exception of the unison) are less

informative than the voice leading itself.

Fig. 4. Quantum RoboSound no. 1. Audio: http://tiny.cc/w0touz

In the second example, we show a perfect convergence to

the target (reward = 0.99), see the score in Figure 5.

Fig. 5. Quantum RoboSound no. 2. Audio: http://tiny.cc/y0touz

B. Soniﬁcation of 3D-motion

To uplift our soniﬁcation to the third dimension, we

include octave change to give an idea of height variation. For

the sake of simplicity, we divided the [0,1] interval along the

z-axis in two parts. Thus, positions in the upper half-space

are soniﬁed with the same notes as in the 2D case, while

positions in the lower subspace contain one-octave lower

sounds, see Figure 3 (b). Octave concatenation is described

in Figure 3 (c) [43]. The musical instruments chosen for the

lower subspace are tenor saxophone, trombone, and double

bass, allowing for lower sounds. R1is thus indicated through

brass (trumpet and trombone), R2with wind instruments

(ﬂute and sax), and R3through strings (cello and double

bass). The two instruments for each robot present timbre

similarity. In the future, we can use octaves of the same

instrument. In our 3D scenario (video http://tiny.cc/

9ctouz), the starting point is chosen near to the surface, in

a point corresponding to A], and the target is in the abyss, at

F]of the lower octave. In this way, the descending motion of

robots is exempliﬁed by a descending pitch. Let us consider

an example of target convergence (Figure 6) and of off-target

convergence (Figure 7).

V. DISCUSSION AND CONCLUSIONS

We presented here a soniﬁcation strategy of a robotic

swarm 2D and 3D motion. Robotic motion is the result

of peer communication and pairwise interactions modeled

upon quantum circuits. Here, the results of soniﬁcation

are sequences of musical chords, providing information on

robotic behavioral patterns. The importance of counterpoint

and voice leading are stressed in contemporary mathematical

studies [15], [44]. Movement perception is naturally (and not

only culturally) associated with sounds: e.g., the Doppler

Fig. 6. Quantum RoboSound no. 3. Audio: http://tiny.cc/11touz

Fig. 7. Quantum RoboSound no. 4. Audio: http://tiny.cc/t0touz

effect explains pitch and loudness variation of a moving

audio source.

Auditory feedback can thus contribute to robotic develop-

ment and motion understanding. In this research, we focused

on ideal robots. As a soniﬁcation-validation strategy, listeners

can be asked to describe their perception of movement. Next

research will extend the proposed approach to real devices. In

the case of a 3D underwater scenario, each robot can transmit

an electric signal that, decoded onboard on the ship, can be

converted into sound and listened by researchers. Practical

implementations will also require charge strategies and real

reward evaluation from robotic sensors (e.g., luminosity

evaluation, shape recognition, sonar and radar applications,

or infrared sensors).

Each algorithmic procedure to move back and forth be-

tween the starting point and the target can lead to charac-

teristic voice leadings, as equivalence classes. Thus, if we

are able to convert the position signal into sound even in the

case of an unknown driving algorithm, sound can help us

retrieve information about it.

Further research will involve a generalization to N robots

with obstacles in their path. Modeling and implementing

robots is ultimately a way to learn something about humans

themselves and their interaction with the environment. Music

can be a privileged way in such an endeavor.

REFERENCES

[1] C. H. Hemelrijk and H. Hildenbrandt, “Schools of ﬁsh and ﬂocks

of birds: Their shape and internal structure by self-organization,”

Interface Focus, vol. 2, pp. 726–737, 2012.

[2] J. Delcourt, N. Bode, and M. Den¨

oel, “Collective Vortex Behaviors:

Diversity, Proximate, and Ultimate Causes of Circular Animal Group

Movements,” The Quaterly Review of Biology, vol. 91, no. 1, pp. 1–24,

2016.

[3] C. Noirot and J. P. E. C. Darlington, “Termite nests: Architecture,

regulation and defence,” in Termites: Evolution, Sociality, Symbioses,

Ecology, T. Abe, D. E. Bignell, and M. Higashi, Eds. Dordrecht:

Springer Netherlands, 2000, pp. 121–139. [Online]. Available:

https://doi.org/10.1007/978-94-017-3223- 9 6

[4] I. Slakov, C. Carrillo-Zapata, D. Jansson, H. Kaandorps, and J. Sharpe,

“Morphogenesis in robot swarms,” Science Robotics, vol. 3, no. 25,

2018.

[5] R. Eberhart, Y. Shi, and J. Kennedy, Swarm Intelligence, ser. The

Morgan Kaufmann Series in Artiﬁcial Intelligence. Burlington, MA,

USA: Morgan Kaufman, 2001.

[6] H. Hamann, Swarm Robotics: A Formal Approach. Cham: Springer,

2018.

[7] E. S¸ahin, “Swarm robotics: From sources of inspiration to domains

of application,” in International Workshop on Swarm Robotics, ser.

Lecture Notes in Computer Science, E. S¸ahin and W. M. Spears, Eds.,

vol. 3342. Berlin/Heidelberg: Springer, 2004.

[8] K. Zhu and M. Jiang, “Quantum Artiﬁcial Fish Swarm Algorithm,”

in Proceedings of the 8th World Congress on Intelligent Control and

Automation, 2010.

[9] X. Dong and M. Sitti, “Controlling two-dimensional collective

formation and cooperative behavior of magnetic microrobot swarms,”

The International Journal of Robotic Research, vol. 39, no. 5,

p. eabe4385, 2020. [Online]. Available: https://journals.sagepub.com/

doi/full/10.1177/0278364920903107

[10] F. Zambonelli, N. Bicocchi, G. Cabri, L. Leonardi, and M. Puviani,

“On Self-adaptation, Self-expression, and Self-awareness in Auto-

nomic Service Component Ensembles,” in 2011 Fifth IEEE Conference

on Self-Adaptive and Self-Organizing Systems Workshops, 2011.

[11] A. S. Wu, R. P. Wiegand, and R. R. Pradhan, “Response probability

enhances robustness in decentralized threshold-based robotic swarms,”

Swarm Intelligence, vol. 14, pp. 233–258, 2020. [Online]. Available:

https://par.nsf.gov/servlets/purl/10182080

[12] M. Salimi and P. Pasquier, “Exploiting Swarm Aesthetics in

Sound Art,” in Proceedings of Art Machines 2: International

Symposium on Machine Learning and Art 2021, School

of Creative Media, City University of Hong Kong, 2021.

[Online]. Available: https://metacreation.net/wp-content/uploads/2021/

06/Exploiting-Swarm- Aesthetics-in- Sound-Art.pdf

[13] F. Santos, “Musical abstractions for multi-robot coor-

dination,” Ph.D. dissertation, 2016. [Online]. Available:

Musicalabstractionsformulti-robotcoordination

[14] J. McLurkin, “The Swarm Orchestra: Temporal Synchronization and

Spatial Division of Labor for Large Swarms of Autonomous Robots,”

online, pp. 1–12, 2022. [Online]. Available: https://groups.csail.mit.

edu/mac/projects/amorphous/6.978/ﬁnal-papers/jamesm- ﬁnal.pdf

[15] D. Tymoczko, “In Quest of Musical Vectors,” in Lecture Notes Series,

Institute for Mathematical Sciences, ser. Mathemusical Conversations,

National University of Singapore, 2016, pp. 256–282.

[16] M. Mannone, V. Seidita, and A. Chella, “Categories, Quantum

Computing, and Swarm Robotics: A Case Study,” Mathematics,

vol. 10, no. 3, p. 372, 2022. [Online]. Available: https://doi.org/10.

3390/math10030372

[17] S. MacLane, Categories for the Working Mathematician. New York:

Springer, 1978.

[18] M. Grandis, Higher Category Theory. Singapore: World Scientiﬁc,

2020.

[19] J. Stolze and D. Suter, Quantum Computing: A Short Course from

Theory to Experiment. Weinheim, Germany: Wiley, 2004.

[20] J. Preskill, “Quantum computing 40 years later,” in Feynman Lectures

on Computation, 2nd ed., A. Hey, Ed. Abingdon, UK: Taylor &

Francis Group, 2021.

[21] A. Wichert, Principles of Quantum Artiﬁcial Intelligence. Singapore:

World Scientiﬁc, 2020.

[22] P. Benioff, “Quantum robots and environments,” Physical Review A,

vol. 58, p. 893, 1998.

[23] V. Ivancevic, “Entangled swarm intelligence: Quantum computa-

tion for swarm robotics,” Mathematics in Engineering, Science and

Aerospace, vol. 7, no. 3, pp. 441–451, 2016.

[24] C. Petschnigg, M. Brandst¨

otter, and H. Pichler, “Quantum Compu-

tation in Robotic Science and Applications,” in IEEE International

Conference on Robotics and Automation (ICRA), 2019.

[25] L. Lamata, M. Quadrelli, C. de Silva, P. Kumar, G. Kanter,

M. Ghazinejad, and F. Khoshnoud, “Quantum Mechatronics,” Elec-

tronics, vol. 10, p. 2483, 2021.

[26] P. Atchade-Adelomou, P. Alonso-Linaje, J. Albo-Canals, and

D. Casado-Fauli, “qRobot: A Quantum Computing Approach

in Mobile Robot Order Picking and Batching Problem Solver

Optimization,” Algorithms, vol. 14, no. 194, 2021. [Online].

Available: https://www.mdpi.com/1999-4893/14/7/194

[27] D. Dong, C. Chen, C. Zhang, and C. Chen, “Quantum robot: structure,

algorithms and applications,” Robotica, vol. 4, pp. 513–521, 2006.

[28] M. Alvarez-Alvarado, F. Alban-Chac ´

on, E. Lamilla-Rubio,

C. Rodr´

ıguez-Gallegos, and W. Vel ´

asquez, “Three novel

quantum-inspired swarm optimization algorithms using different

bounded potential ﬁelds,” Nature Scientiﬁc Reports, vol. 11, p.

11655, 2021. [Online]. Available: https://www.nature.com/articles/

s41598-021- 90847-7

[29] A. Koukam, A. Abbas-Turki, V. Hilaire, , and Y. Ruichek, “Towards

a Quantum Modeling Approach to Reactive Agents,” in 2021 IEEE

International Conference on Quantum Computing and Engineering

(QCE), 2021.

[30] S. N˘

ad˘

aban, “From classical logic to fuzzy logic and quantum logic:

a general view,” International Journal of Computers Communications

and Control, vol. 16, no. 1, p. 4125, 2021.

[31] A. Sabra and W. Fung, “A Fuzzy Cooperative Localisation Framework

for Underwater Robotic Swarms,” Sensors, vol. 20, p. 5496, 2020.

[32] M. Dorigo, G. Theraulaz, and V. Trianni, “Reﬂections on the future of

swarm robotics,” Science Robotics, vol. 5, no. 49, p. eabe4385, 2020.

[33] D. Dong, C. Chen, H. Li, and T.-J. Tarn, “Quantum Reinforcement

Learning,” in IEEE Transactions on Systems Man and Cybernetics

Part B (Cybernetics), vol. 38, no. 5, 2008. [Online]. Available:

https://fr.art1lib.org/book/18461932/6b4359

[34] Y. Kwak, W. J. Yun, S. Jung, J.-K. Kim, and J. Kim, “Introduction

to Quantum Reinforcement Learning: Theory and PennyLane-based

Implementation,” in International Conference on Information and

Communication Technology Convergence (ICTC), 2021.

[35] M. Mannone, V. Seidita, and A. Chella, “Modeling and Designing a

Robotic Swarm: a Quantum Computing Approach,” 2022.

[36] L. Pitonakova, R. Crowder, and S. Bullock, “Task Allocation

in Foraging Robot Swarms: The Role of Information Sharing ,”

Artiﬁcial Life Conference Proceedings: ALIFE 2016, the Fifteenth

International Conference on the Synthesis and Simulation of

Living Systems, vol. 17, pp. 93–105, 2020. [Online]. Available:

https://direct.mit.edu/isal/proceedings/alif2016/28/306/99491

[37] S. Berman, Q. Lindsey, M. S. Sakar, V. Kumar, and S. C. Pratt,

“Experimental Study and Modeling of Group Retrieval in Ants as

an Approach to Collective Transport in Swarm Robotic Systems,”

Proceedings of the IEEE, vol. 99, no. 9, pp. 1470–1481, 2011.

[38] W.-M.-. Shen, P. Will, A. Galstyan, and C.-M. Chuong, “Hormone-

Inspired Self-Organization and Distributed Control of Robotic

Swarms ,” Autonomous Robots, vol. 17, pp. 93–105, 2020. [Online].

Available: http://www.agentgroup.unimo.it/Zambonelli/didattica/cas/

L14/Shen RobotSwarms.pdf

[39] A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical

Description of Physical Reality Be Considered Complete?” Physical

Review Letters, vol. 47, p. 777, 1935. [Online]. Available:

https://journals.aps.org/pr/pdf/10.1103/PhysRev.47.777

[40] J. S. Bell, “On the Einstein Podolsky Rosen Paradox,” Physics,

vol. 1, no. 3, pp. 195–200, 1964. [Online]. Available: https:

//journals.aps.org/ppf/pdf/10.1103/PhysicsPhysiqueFizika.1.195

[41] R. Jozsa and N. Linden, “On the Role of Entanglement in

Quantum-Computational Speed-Up,” in Proceedings: Mathematical,

Physical and Engineering Sciences, vol. 459, no. 2036, 2003, pp.

2011–2032. [Online]. Available: https://www.jstor.org/stable/3560059

[42] D. M. Greenberger, M. A. Horne, and A. Zeilinger, “Going Beyond

Bell’s Theorem,” in Bell’s Theorem, Quantum Theory, and Concep-

tions of the Universe, M. Kafatos, Ed. Kluwer: Dordrecht, 1989, pp.

69–72.

[43] F. Lerdhal, “Tonal Pitch Space,” Music Perception, vol. 5, no. 3, pp.

315–350, 1998.

[44] D. Tymoczko, “The Geometry of Musical Chords,” Science, vol. 313,

2006.