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Citation: Mannone, M.; Valeria S.;
Antonio, C. Categories, Quantum
Computing, and Swarm Robotics:
A Case Study. Mathematics 2022,10,
372. https://doi.org/10.3390/math
10030372
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mathematics
Article
Categories, Quantum Computing, and Swarm Robotics:
A Case Study
Maria Mannone 1,2,3,* , Valeria Seidita 1and Antonio Chella 1,4
1Department of Engineering, University of Palermo, 90128 Palermo, Italy; valeria.seidita@unipa.it (V.S.);
antonio.chella@unipa.it (A.C.)
2Department of Mathematics and Computer Sciences, University of Palermo, 90123 Palermo, Italy
3
European Centre for Living Technology (ECLT), Dipartimento di Scienze Ambientali, Informatica e Statistica
(DAIS), Ca’ Foscari University of Venice, 30172 Venice, Italy
4ICAR-CNR National Research Council, 90146 Palermo, Italy
*Correspondence: mariacaterina.mannone@unipa.it or maria.mannone@unive.it
Abstract:
The swarms of robots are examples of artificial collective intelligence, with simple individ-
ual autonomous behavior and emerging swarm effect to accomplish even complex tasks. Modeling
approaches for robotic swarm development is one of the main challenges in this field of research.
Here, we present a robot-instantiated theoretical framework and a quantitative worked-out example.
Aiming to build up a general model, we first sketch a diagrammatic classification of swarms relating
ideal swarms to existing implementations, inspired by category theory. Then, we propose a matrix
representation to relate local and global behaviors in a swarm, with diagonal sub-matrices describing
individual features and off-diagonal sub-matrices as pairwise interaction terms. Thus, we attempt to
shape the structure of such an interaction term, using language and tools of quantum computing for a
quantitative simulation of a toy model. We choose quantum computing because of its computational
efficiency. This case study can shed light on potentialities of quantum computing in the realm of
swarm robotics, leaving room for progressive enrichment and refinement.
Keywords: swarm robotics; quantum computing; 4-qubit system; matrix representation; colimit
1. Introduction
Acrobatics evolution of flocks of birds in our skies and colorful movements of schools
of fish in our oceans inspire poets and artists, as well as computer scientists and engineers.
Complex behaviors exhibited by swarms of animals [
1
,
2
], in fact, often inspire the develop-
ment of swarms of robots, passing through toy models which are progressively enriched
and refined [3].
In a swarm of robots [
3
–
5
], every single robot is performing simple tasks and showing
a simple behavior, but the interaction and information exchange amongst robots allows
the accomplishment of more complex tasks, impossible to be achieved by a single unit.
Typically, the collaboration in a swarm is decentralized: there is no such a thing as a “robot
leader;” instead, every single robot is acting autonomously, reacting to the information
received by its neighbors, and transmitting information about its own activity. Swarms
of robots show the emergence of global behavior and a form of collective intelligence—a
classic example is given by ants’ behavior. Thus, we can talk of swarm intelligence [
6
]. Exam-
ples of swarm intelligence are stochastic diffusion search [
7
], ant colony optimization [
8
],
and artificial swarm intelligence [9].
A swarm is an example of a complex system, which can be instantiated as a robot
swarm. Applications of swarm robotics include field tasks, underwater tasks, and air tasks.
Terrestrial, underwater, and flying autonomous simple vehicles have thus to be developed
and made communicating between them. There are also miniature robots going inside the
human body for healthcare purposes [10].
Mathematics 2022,10, 372. https://doi.org/10.3390/math10030372 https://www.mdpi.com/journal/mathematics
Mathematics 2022,10, 372 2 of 11
From an engineering point of view, the connection between local behavior (single or
subgroups of robots in a swarm) and the global behavior of a swarm is not trivial. Some
existing research exploits particle analogies from physics, with a microscopic behavior
described by Langevin equation, and a macroscopic behavior formalized through the
Fokker–Planck Equation [
3
]. A great inspiration source for mathematical modeling is also
given by quantum mechanics and related quantum computing.
Recently, the widespread interest in quantum computing [
11
], motivated by its com-
putational power [
12
] and formal elegance is influencing several fields of research, e.g.,
quantum computing has recently been applied to machine learning for artificial intelli-
gence [
13
]. In a recent study [
14
], fuzzy logic is applied to a swarm of robots. Quantum
logic can be considered as a particular case of fuzzy logic [15].
Some first applications of quantum computing to swarm robotics exploit evolutionary
methods inspired by particle modeling dynamics and involving Monte-Carlo simula-
tions [16], and the strong hypothesis of entanglement between swarm robots [17].
Quantum applications to robots are a flourishing research field. With the exceptions
of some pioneering works, quantum applications to swarm robotics appear instead as a
largely unexplored domain. Another gap is the lack of simple connections of robotics,
and in particular of swarm robotics, with quantum computing at and fundamental level.
A closely-related research [
18
], in fact, focuses on a single agent rather than on a swarm
of robots.
The use of quantum computing is inspired by core ideas and discoveries in basic
quantum mechanics, with energy quantization, state superposition, destructive measure-
ment. Physics Nobel prize Richard Feynman had already hypothesized that the application
of quantum principles to computer science could have enabled the construction of more
powerful computers [
11
]. In quantum computing, the information units 0, 1 are identified
with the ground and excited states of a (hydrogen) atom, respectively. Quantum computing
makes use of reversible logic gates, in analogy with invertible quantum operators. The use
of quantum computing is mainly motivated by computational increased efficiency. This is a
flourishing research area, and thus, a connection with quantum computing can strengthen
other fields of research.
In our contribution, we aim to create a general theoretical approach for interacting
multi-entities with an emerging swarm behavior. As an instance of complex models with
swarm behavior, we focus on robots, defining interaction terms as submatrices of the
swarm matrix. As an example of an interaction term, we create a toy model with two
robots, modeling robots’ behavior through a quantum circuit.
In fact, in this article we aim to address the issue of the connection between macro- and
micro-behaviors in a swarm of robots, crucial for swarm modeling, by using nested matrices
and the basics of quantum computing. While we choose to focus on robots, our method-
ology is meant to be more general: in fact, we model relationships between information
exchange and behavior of interacting simple entities, showing an emergent behavior.
First, we discuss a framework to describe the progressive “embodiment” of the swarm
idea, from an ideal swarm to a particular system.
Second, we propose a swarm description in terms of nested matrices describing single
robots’ behaviors and their interaction. In fact, we can translate the swarm condition
into a matrix structure. While in [
17
] there is a matrix where all elements (robots) are
entangled, here we relax this hypothesis, and we include terms describing autonomous
behavior of robots, distinguishing them from the off-diagonal terms for robots’ interactions.
A swarm can be described by a block matrix, where the sub-matrices along the main
diagonal describe the behavior of isolated robots, and the other sub-matrices indicate the
interactions between them.
Third, we work out a simple example with two robots, formalized through four
quantum qubits (where the qubit is the quantum information unit), discussing the ob-
tained results.
Mathematics 2022,10, 372 3 of 11
The experiments with the quantum circuit (corresponding to the truth table of Table 1)
are set up by typing the provided code (Section 2) on IBM Quantum Composer and running
it through a simulator (e.g., QASM simulator) or an IBM-owned quantum computer (e.g.,
the computer in Bogotà), to which the circuit has been remotely sent. We initialize input
qubits through suitable quantum gates (Hadamard, Not, and Ry in our study). Each
experiment consists of 1024 shots of the code. The population for each output state is
indicated in Table 2.
The article is organized as follows. In Section 1.1, we summarize the key concepts
of swarm robotics. In Section 2, we provide details on the mathematical formalization
of the core idea, introducing the 4-qubit case study. In Section 3, we present our results.
In Section 4, we summarize our study and envisage possible research developments.
1.1. Swarms of Robots
Swarm robotics is an application of swarm intelligence [
6
,
19
] to robotics. The core
idea of this discipline is the reference to social insects behavior. In fact, robots in a swarm
are meant to autonomously coordinate to achieve complex goals. Organisms in a natural
swarm use simple rules to govern their behavior. Every single organism in a swarm can
perform a few simple actions. Communication and interaction with other organisms in the
swarm enable the achievement of complex goals, e.g., heavy prey transporting, foraging,
or massive and complex structures building. An intuitive definition of swarm robotics is
the following: swarm robotics is the study of how a large number of relatively simple physically
embodied agents can be designed such that a desired collective behavior emerges from the local
interactions among agents and between the agents and the environment [20].
A swarm of robots is constituted by a set of interacting robots, individually performing
a simple job, and collectively achieving a complex task [
3
,
4
]. A complex task is achieved
through task partitioning. Swarms of robots show self-organizational properties. The swarm
behavior is considered as an emerging property. A swarm of robots is characterized by
redundancy (the lack of a unity connection does not affect the whole behavior), flexibility
(no specialization is required), and scalability (the control algorithms do not depend upon
the size of the swarm). In swarm robotics, we can distinguish between a micro level and
amacro level. The micro-level concerns the local behavior, describing single unities and
pairwise interactions. The macro-level concerns global behavior, that is, the behavior of the
whole swarm [3].
1.2. Ideal Swarms
In Section 2, we introduce the concept of an ideal swarm of robots. The ideal swarm is
ontologically different from any feasible swarm: it is an abstraction. It can be visualized as
a swarm of robots that:
• 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).
It is impossible to envisage such a swarm or conceive any similar implementation.
In a nutshell, swarm conditions are relationships between individual behaviors of
single robots and the overall actions of the swarm. Complex swarm behavior is supposed to
emerge from simple individual actions and, as in our model, simple pairwise interactions.
Differences in modeling between swimming [
21
], flying [
22
], and walking swarms [
23
]
intuitively reside on locomotion features and target scenarios to which robots should
be adapted.
Mathematics 2022,10, 372 4 of 11
2. Theoretical Framework and Methods
In this article, swarm conditions become conditions on matrices. Let us consider an
ideal swarm (Section 1.2); we can call it
S
. It can be obtained as an idealization from a
general yet feasible swarm structure; let us call it
S
. Such a general swarm can be specialized
into main typologies of swarms, such as swimming swarms,flying swarms, and walking swarms.
Each one of these three classes can be distinguished into more and more specific swarms,
toward existing swarms, made of simple and cheap robots, which communicate between
them and collaborate to perform tasks.
S(ideal swarm)
S(general swarm)
Sf lyin g Sswimmi ng
Sf
ASf
BSf
CSs
ASs
BSs
C
!
ff
lf→s
gf
fs
gs
sf
A
ff
A
hf
A,B
F
sf
B
hf
B,C
F
Sf
C
F
ss
A
hs
A,B
ss
B
hs
B,C
ss
C
fs
C
(1)
Moving downward in the diagram of Equation
(1)
we get a progressive embodiment
of the general idea, toward specific realizations of swarms of robots. Vice versa, the upward
paths leads to a progressive abstraction. This diagram can be compared with the species
classification in biology, which has been formalized through the concept of colimit in Cate-
gory Theory [
24
]. In [
25
], emergence of species similarities are shown through convergence
of arrows. Here, emergence of similarities in swarm behaviors can also be thought of as
arrows convergence. In fact, diagram
(1)
is shaped as a colimit construction. We can have
emergent behavior similarities while comparing swarms between them, but also regarding
the very definition of a swarm itself: the swarm behavior is more than the sum of single
robots’ movements.
In diagram
(1)
, horizontal comparisons are comparisons across swarms with the
same degree of abstraction. The comparison between two swarms
Sf
A
,
Sf
B
belonging to
the category flying swarms is performed through suitable morphisms
hA,B:Sf
A→Sf
B
.
The comparison between characteristics of the category of flying robots,
Sf lyi ng
, with the
characteristics of the category of swimming robots,
Sswimming
, is represented by the arrow
lf→s:Sf lyi ng →Sswimmin g
. Letter
F
between specific swarms of a category flying and
swarms of a category swimming indicates a functor (a generalization of functions, mapping
points of a category into points another category, and morphisms of a category into mor-
phisms of another category). While it is not indicated here for reasons of graphical clarity,
Falso maps hf
A,B→hs
A,B,hf
B,C→hs
B,C, and hf
A,C→hs
A,C.
Given an ideal swarm
S
, here treated as a categorical colimit, there must be some
characteristics to be taken into account to create a general yet feasible swarm
S
. This is
the meaning of the dotted arrow. According to the definition of a colimit in Category
Theory, there is a unique morphism
! : S → S
such that
fx=!·gx
, where
x
is the label of
a swarm category and
·
indicates the composition; e.g.,
ff=!·gf
. Specific swarms can
be related to the abstract swarms through arrow compositions; e.g.,
gf·sf
A:Sf
A→ S
and
ff·sf
A:Sf
A→S.
Each swarm in diagram
(1)
can in principle be described by a matrix. In the case of
existing physical swarms (
Sf
A
,
. . .
,
Sf
C
), all elements of matrices can be computed. Let us
now focus on a generic swarm constituted by
n
robots. The block matrix describing it is
sketched in Equation
(2)
, where the sub-matrices along the principal diagonal describe
the behavior of each single robot
R1
,
. . . Rn
, and the other sub-matrices are the pairwise
Mathematics 2022,10, 372 5 of 11
interaction terms. If these interaction terms are null matrices, there are no swarm effects.
Thus, off-diagonal matrices can remind us of the coherences in quantum mechanics, i.e., the
off-diagonal terms representing entanglement.
Sn=
R1R1∗R2. . . R1∗Rn−1R1∗Rn
R2∗R1R2. . . R2∗Rn−1R2∗Rn
.
.
.
Rn−1∗R1Rn−1∗R2. . . Rn−1Rn−1
Rn∗R1Rn∗R2. . . Rn∗Rn−1Rn
(2)
If there are only n=2 robots, the matrix is simplified as shown in Equation (3):
S2=R1R1∗R2
R2∗R1R2(3)
The matrix S2is a block matrix which can be obtained as in Equation (4):
S2=1 0
0 0⊗R1+0 1
0 0⊗(R1∗R2) + 0 0
1 0⊗(R2∗R1) + 0 0
0 1⊗R2, (4)
where
⊗
is the Kronecker product (tensor product) between matrices. The matrices
R1
,
R2
describe the behavior of the two isolated robots. The interaction terms
R1∗R2
and
R2∗R1
are not equal, because they depend on:
• Signals transmitted from R1to R2and reactions of R2(term R1∗R2),
• Signals transmitted from R2to R1and reactions of R1(term R2∗R1), respectively.
How are the interaction terms built? For
R1∗R2
, we may consider the signal robot 1
is transmitting to robot 2, and the reaction of the last one. A possible communication is
“where I am, where I go.” However, even in a toy model for a swarm it is important to define
a task. A task could be, for example, the achievement of a goal, as a target to reach, or as a
reward to obtain. (In fact, target search is a classic task for a swarm of robots [
26
]). Thus,
in our simple description the signal from a robot provides information on its position as a
portion of a linear space and its target achievement—and thus, the content of the message
would be “where I am, what I found.” Hereinafter, to take into account an inner uncertainty
related to the decision-making autonomy of robots, we formulate a probabilistic approach
based on quantum state superposition. Thus, the possible communication scheme from
R1
to R2could be a matrix as:
place down reward no
place up reward yes,
where each entry is the probability amplitude to have that signal. To enrich this idea, we
can add the reaction by R2:
(R1∗R2)(t) =
R1place down(t0)R1reward no (t0)
R1place up (t0)R1reward yes (t0)
−R2place down (t1)
−R2place up (t1)
, (5)
where, to a signal from
R1
at time
t0
, corresponds a position displacement by
R2
at time
t1
.
The empty entries correspond to the reward from
R2
, that will be made known by the robot
only at
t2
. The term
R2∗R1
will thus have the same structure. To date, we did not choose a
specific framework, because we are interested in creating a general approach, which can be
instantiated with simple, interacting robots. In the proposed toy model, we assume to have
two simple robots moving back and forth along a line.
Mathematics 2022,10, 372 6 of 11
Let us now work out a quantitative example. To this aim, we can treat the behavior of
each robot as a 2-qubit quantum state, where, for the sake of simplicity, each qubit indi-
cates the superposition of possible states of one-dimensional position (up
=
1,
down =0
)
and of “rewards” as a reached target (yes
=
1, no
=
0). In total, we have a system with
four qubits, that we used for a simulation with IBM simulators and real quantum com-
puters. (https://www.ibm.com/quantum-computing/ accessed on 10 December 2021).
In Equation
(6)
, we indicate probability amplitudes to obtain a specific measure outcome
as αi,βi,γi,δi,i=1, 2.
R1→ |q0(t)i=α1(t)|0i+β1(t)|1i,|q1(t)i=γ1(t)|0i+δ1(t)|1i
R2→ |q2(t)i=α2(t)|0i+β2(t)|1i,|q3(t)i=γ2(t)|0i+δ2(t)|1i(6)
The matrix of Equation (5) will then be:
(R1∗R2)(t) =
α1(t0)γ1(t0)
β1(t0)δ1(t0)
−α2(t1)
−β2(t1)
. (7)
Let us analyze a possible realization of the scheme of communication/reaction building a
reversible logic gate. The logic behind the truth tables in Table 1is the following. At time
t0
, if
R1
is in position up (
q0=
1) and it gets the reward (
q1=
1), then
R2
reaches it at
t1
,
independently from its former position. However, if
R1
does not get the reward (
q1=
0),
then
R2
explores the space down (
q2=
0). Same reasoning if the starting position is the
inverse. Then, the roles of
R1
and
R2
are exchanged at
t2
and
t3
. (Time in quantum gates is
simultaneous, while we know that a robot, received the communication from the other one,
implements the decision to change the position or keep it in the following time instant).
The permutation matrix associated with Table 1is presented in Equation
(8)
, with quantum
states ordered as 00, 01, 10,11.
Table 1.
Truth tables (reversible equivalents of XNOR gates), representing the interaction between
robot 1,
R1
(
q0
: position,
q1
: reward) and robot 2,
R2
(
q2
: position,
q3
: reward). At the beginning (left
table),
R1
communicates its position (down/up) and reward (no/yes); then it waits, and, according to
its information,
R2
reaches
R1
or not (when
R2
reaches
R1
,
q2
becomes equal to
q0
). Then (right table),
we have the symmetric situation, where
R2
communicates position and reward, and
R1
decides to
reach it nor not (if
R1
reaches
R2
,
q0
becomes equal to
q2
). Equation
(8)
shows the corresponding
permutation matrix.
Gate t0→t1Gate t1→t2
input output input output
q0q1q0q2q2q3q2q0
00010001
01000100
10101010
11111111
0100
1000
0010
0001
(8)
The matrix in Equation
(9)
shows the Toffoli gate, exploited to implement our proposed
gate. The Toffoli gate is a reversible quantum gate, which takes two inputs and gives one
output. It is often indicated as ccx, that is, a not (x) with two conditions. Figure 1shows the
considered quantum circuit.
Mathematics 2022,10, 372 7 of 11
1000
0100
0001
0010
(9)
Figure 1.
The considered quantum circuit for
|q0i=1
√2|
0
i+1
√2|
1
i
,
|q1i=q2
3|
0
i+1
√3|
1
i
. Chang-
ing the initializations of q0and q1through suitable Rygates, we can span the different cases.
In Section 3, we show our results, obtained through the interface IBM Quantum
Composer (https://quantum-computing.ibm.com/composer/ accessed on 10 December
2021), an open computing environment, with simulators and quantum computers which
can be accessed remotely. Real IBM quantum computers allow up to five qubits.
Because it was not possible to change probabilities directly in Quantum Composer,
we created the Qiskit (https://qiskit.org/ accessed on 10 December 2021) code and then
exported the code lines in QASM, the coding environment required for Quantum Composer.
In this way, each specific probability amplitude was obtained as a special quantum gate
Ry
;
e.g., initial_state = [0.5, 0.5] (that is,
1
√2|
0
i+1
√2|
1
i
) was written as
Ry(π
2)
. This particular
state can also be obtained with a Hadamard gate acting on
|
0
i
. In the following, we present
our code. The state
|
1
i
is obtained through a Not gate as
x|
0
i
, because states in Quantum
Composer are initialized to 0 by default. In the code, if both
|q0i
and
|q1i
are
|
0
i
(indicating
position 0 and failure to reach the target, respectively), then
|q2i
is flipped to
|
1
i
, that is,
R2
goes to position 1. If both
|q0i
and
|q1i
are
|
1
i
(indicating position 1 and success in reaching
the target, respectively), then
|q2i
is flipped to
|
1
i
as well. This second effect is obtained by
adding Not gates (x) before qubits
|q0i
and
|q1i
. If
|q0i
and
|q1i
are different between them,
then |q2iremains in the default state |0i.
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];
Mathematics 2022,10, 372 8 of 11
3. Results
The dataset used in the experiment is summarized in Table 2. To replicate the ex-
periment, it is sufficient to create an account on IBM Quantum (https://www.ibm.com/
quantum-computing/ accessed on 10 December 2021), open a Quantum Composer new
file, type the given code, and customize the initial states according to the
|q0i
,
|q1i
entries
in Table 1. In IBM Quantum Composer, states are initialized by default as
|
0
i
. To have
|
1
i
,
it is sufficient to use a NOT gate:
x|
0
i=|
1
i
. To have the 1
/√2
superposition of 0 and 1,
one can use the Hadamard gate H or the
Ry(π/
2
)
gate. To obtain the other combinations,
it is necessary to opportunely change the angle of the
Ry(θ)
gate. Then, one can hit the
“Setup and Run” command, selecting one of the quantum computers available or one of the
simulators, and waiting in a queue for the required time. The obtained values can slightly
fluctuate because of quantum error, and because of the very nature of quantum world:
probabilistic. No other parameters are required in this experiment. We tested the following
combinations of initial states:
|q0i=|1i,|q1i=|1i;
|q0i=|1i,|q1i=|0i;
|q0i=1
√2|0i+1
√2|1i,|q1i=|0i;
|q0i=1
√2|0i+1
√2|1i,|q1i=|1i;
|q0i=|0i,|q1i=r2
3|0i+1
√3|1i;
|q0i=|1i,|q1i=r2
3|0i+r2
3|1i;
|q0i=|0i,|q1i=1
√3|0i+r2
3|1i;
|q0i=|1i,|q1i=1
√3|0i+r2
3|1i;
|q0i=1
√2|0i+1
√2|1i,|q1i=r2
3|0i+1
√3|1i;
|q0i=1
√2|0i+1
√2|1i,|q1i=1
√3|0i+r2
3|1i.
(10)
In Table 2, we compare the outcomes obtained with IBM simulators and IBM quantum
computers. With the quantum computer, the results are mostly as expected, but the noise is
high; the simulators give a result that perfectly matches theoretical expectations.
A case of particular interest are the eigenstates, to measure the amount of possible
quantum noise, and the case with random position of
R1
(
|q0i
with 50% of chances to be
in 0 and to be in 1), and maximal certitude on the reward information (
|q1i
with 100% of
chances to be in 1 and 0% to be in 0, or vice versa).
Results with real quantum computers can differ according to the chosen machine,
and they present a higher noise. The noise is particularly evident when we tested
q0
,
q1
as eigenstates. The simulator gave us all the outcomes of
q2
as 0 or as 1, as expected from
Table 1
, while the quantum computer also presented a non-zero population of the unexpected
outcome. Each run of the code gave 1024 shots of measurements. Fluctuations at each run of
the code, not displayed here, are present for both typologies of computing devices.
It is not possible to carry out a comparison with other methods, because the proposed
methodology is new. To the best of our knowledge, the only case of a close-related research
is the application proposed by Koukam et al. [
18
], where an IBM Quantum computer has
been used for a reactive agent. In their study, a single robot (and not a swarm) modifies its
behavior according to the external information recovered through suitable sensors.
Mathematics 2022,10, 372 9 of 11
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.
|q0i |q1i |q2i(Expected) Counts for 0, 1 Device
|1i |1i1 0; 1024 S
455; 569 C
|1i |0i0 1024; 0 S
527; 497 C
1
√2|0i+1
√2|1i|1iuncertain 494; 530 S
502; 522 C
1
√2|0i+1
√2|1i|0iuncertain 527; 497 S
431; 593 C
|0iq2
3|0i+1
√3|1i0 686; 338 S
531; 493 C
|1iq2
3|0i+1
√3|1i0 702; 322 S
570; 414 C
|0i1
√3|0i+q2
3|1i1 315; 709 S
452; 572 C
|1i1
√3|0i+q2
3|1i1 335; 669 S
376; 648 C
1
√2|0i+1
√2|1iq2
3|0i+1
√3|1iuncertain 518; 506 S
351; 673 C
1
√2|0i+1
√2|1i1
√3|0i+q2
3|1iuncertain 508; 516 S
433; 591 C
4. Discussion and Conclusions
In this article, we started from theoretical considerations on swarms of robots, from
ideal, abstract swarms down to typologies of swarms with specific existing realizations.
We proposed a matrix description of a generic swarm, trying to connect the action of each
single robot with the overall behavior. The proposed swarm matrix is a block matrix with
sub-matrices indicating the motion of single robots, and off-diagonal blocks representing
pairwise interaction terms, with signals from the
i
-th robot and behavioral response for the
j
-th robot. Aiming to shape an example of these interaction terms, we exploit the basics of
quantum computing to work out a toy model with four qubits, implementing it through
IBM computational resources and analyzing the obtained results.
The presented results provide information regarding output states’ likelihood. This
information is related to the decision-making outcomes in a toy 2-robot swarm. The input
represents incoming information from the first robot, and the output indicates the behav-
ioral response from the second robot. The proposed quantum circuit acts as an overall
gate, corresponding to an interaction term at time
t
in the proposed matrix representa-
tion. The interaction terms model local behaviors related to global behaviors through the
proposed matrix.
Let us briefly analyze the pros and cons of the proposed strategy. The cons are
mainly related to technological limitations: IBM Quantum computers are freely available
only up to five qubits, while IBM Quantum simulators are available up to 32, 63, 100
and, very recently (after the submission of this article), 5000 qubits. The pros include
computational power of quantum computing, proved for classic problems such as integer
factorization, and generalization power of the proposed theoretical framework. In fact, we
can consider robot interactions as instances of social behaviors, be they of people, animals,
Mathematics 2022,10, 372 10 of 11
or neuronal entities. Thus, the proposed methodology is not limited to robots: it can instead
be applied to interactions amongst multiple entities, constituting complex systems with an
emergent behavior.
Future developments of the proposed research will involve the construction of more
complex models, with a higher number of qubits, describing more degrees of freedom
of robots. From the conceptual level, additional degrees of freedom can help us model
more and more complex group behaviors, also imitating aspects of natural swarm scenar-
ios [
1
,
2
]. Another point to be addressed in future research is map building as an effect of
memory where robots explore their environment looking for the target. Further research
can draw upon existing studies on agent-based decision-making [
27
], localization and map
building developed for single [
28
] and multiple robots [
29
,
30
], to build a theoretical unifica-
tion and envisage a quantum perspective on swarms, also exploiting quantum computing
computational resources.
Author Contributions:
Conceptualization, M.M., V.S., and A.C.; methodology, M.M. and V.S.; soft-
ware, M.M.; validation, M.M.; formal analysis, M.M.; investigation, M.M., V.S. and A.C.; resources,
A.C.; data curation, M.M., V.S. and A.C.; writing—original draft preparation, M.M.; writing—review
and editing, M.M., V.S. and A.C.; visualization, M.M.; supervision, V.S. and A.C.; project adminis-
tration, A.C. and V.S.; funding acquisition, A.C. and V.S. All authors have read and agreed to the
published version of the manuscript.
Funding:
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.
Data Availability Statement:
The data presented in this study are openly available in GitHub:
https://github.com/medusamedusa/quantum-world (accessed on 10 December 2021).
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
Delcourt, J.; Bode, N.W.; Denoöel, M. Collective Vortex Behaviors: Diversity, Proximate, and Ultimate Causes of Circular Animal
Group Movements. Q. Rev. Biol. 2016,91, 1–24. [CrossRef] [PubMed]
2.
Hemelrijk, C.H.; Hildenbrandt, H. Schools of fish and flocks of birds: Their shape and internal structure by self-organization.
Interface Focus 2012,2, 726–737. [CrossRef] [PubMed]
3. Hamann, H. Swarm Robotics: A Formal Approach; Springer: Cham, Switzerland, 2018.
4.
Dorigo, M.; Theraulaz, G.; Trianni, V. Swarm Robotics: Past, Present, and Future [Point of View]. Proc. IEEE
2021
,109, 1152–1165.
Available online: https://hal.archives-ouvertes.fr/hal-03362874/document (accessed on 10 December 2021 ). [CrossRef]
5.
Schranz, M.; Umlauft, M.; Sende, M.; Elmenreich, W. Swarm Robotic Behaviors and Current Applications. Front. Robot. AI
2020
,
7, 36. [CrossRef] [PubMed]
6.
Eberhart, R.; Shi, Y.; Kennedy, J. Swarm Intelligence; The Morgan Kaufmann Series in Artificial Intelligence; Morgan Kaufman:
Burlington, MA, USA, 2001.
7.
Bishop, J.M. Stochastic Searching Networks. In Proceedings of the 1st IEEE Conference Artificial Neural Networks, London, UK,
16–18 October 1989. Available online: http://www.reading.ac.uk/web/files/sse/sds-ssn.pdf (accessed on 10 December 2021 ).
8. Dorigo, M.; Birattari, M.; Stutzle, T. Ant colony optimization. IEEE Comput. Intell. Mag. 2001,4, 28–39.
9.
Rosenberg, J.L. Human Swarms, a real-time method for collective intelligence. In Proceedings of the European Conference on
Artificial Life, York, UK, 20–24 July 2015; pp. 658–659.
10.
Dong, X.; Sitti, M. Controlling two-dimensional collective formation and cooperative behavior of magnetic microrobot swarms.
Int. J. Robot. Res.
2020
,39, 5. Available online: https://journals.sagepub.com/doi/full/10.1177/0278364920903107 (accessed on
10 December 2021 ). [CrossRef]
11.
Preskill, J. Quantum computing 40 years later. In Feynman Lectures on Computation, 2nd ed.; Hey, A.J.G., Ed.; Taylor & Francis
Group: Abingdon, UK, 2021.
12. Stolze, J.; Suter, D. Quantum Computing: A Short Course from Theory to Experiment; Wiley: Weinheim, Germany, 2004.
13. Wichert, A. Principles of Quantum Artificial Intelligence; World Scientific: Singapore, 2020.
14.
Sabra, A.; Fung, W.-K. A Fuzzy Cooperative Localisation Framework for Underwater Robotic Swarms. Sensors
2020
,20, 5496.
[CrossRef]
15.
N˘ad˘aban, S. From Classical Logic to Fuzzy Logic and Quantum Logic: A General View. Int. J. Comput. Commun. Control.
2021
,16,
1841–9836. [CrossRef]
Mathematics 2022,10, 372 11 of 11
16.
Li, Z.; Liu, W.; Li, E.G; Li, L. Path Planning Method for AUV Docking Based on Adaptive Quantum-Behaved Particle Swarm
Optimization. IEEE Access Multidiscip. 2019,7, 78665–78674. [CrossRef]
17. Ivancevic, V.G. Entangled Intelligence: Quantum Computation for swarm robotics. Math. Eng. Sci. Aerosp. 2018,7, 441–451.
18.
Koukam, A.; Abbas-Turki, A.; Hilaire, V.; Ruichek, Y. Towards a Quantum Modeling Approach to Reactive Agents. In Proceedings
of the 2021 IEEE International Conference on Quantum Computing and Engineering (QCE), Broomfield, CO, USA, 18–22 October
2021; pp. 130–136.
19.
Bonabeau, E.; Theraulaz, G.; Dorigo, M. Swarm Intelligence: From Natural to Artificial Systems; Oxford University Press: New York,
NY, USA, 1999; pp. 3–77.
20.
¸Sahin, E. Swarm robotics: From sources of inspiration to domains of application. In International Workshop on Swarm Robotics;
¸Sahin, E., Spears, W.M., Eds.; Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 2004; Volume 3342.
21. Berlinger, F.; Gauci, M.; Nagpal, R. Implicit coordination for 3D underwater collective behaviors in a fish-inspired robot swarm.
Sci. Robot. 2021,6, 50. [CrossRef] [PubMed]
22.
Soria, E.; Schiano, F.; Floreano, D. Predictive control of aerial swarms in cluttered environments. Nat. Mach. Intell.
2021
,3, 545–554.
[CrossRef]
23.
Ozkan-Aydin, Y.; Goldman, D.I. Self-reconfigurable multilegged robot swarms collectively accomplish challenging terradynamic
tasks. Sci. Robot. 2021,6, 56. [CrossRef] [PubMed]
24. Mac Lane, S. Categories for the Working Mathematician; Springer: New York, NY, USA, 1978.
25. Mannone, M. Mathematics, Nature, Art; Palermo University Press: Palermo, Italy, 2019.
26.
Ismail, Z.H.; Hamami, M.G.M. Systematic Literature Review of Swarm Robotics Strategies Applied to Target Search Problem
with Environment Constraints. Appl. Sci. 2021,11, 2383. [CrossRef]
27.
Chella, A.; Lanza, F.; Seidita, V. Decision Process in Human-Agent Interaction: Extending Jason Reasoning Cycle. In Engineering
Multi-Agent Systems; Weyns, D., Mascardi, V., Ricci, A., Eds.; EMAS 2018. Lecture Notes in Computer Science; Springer: Cham,
Switzerland, 2018; Volume 11375, pp. 320–339.
28.
Dissanayake, M.W.M.; Newman, P.; Clark, S.; Durrant-Whyte, H.F.; Csorba, M. A Solution to Simultaneous Localization and Map
Building (SLAM) Problem. IEEE Trans. Robot. Autom. 2001,17, 229–241. [CrossRef]
29.
Zhou, Y.; Hu, H.; Liu, Y.; Lin, S.; Ding, Z. A Real-Time and Fully Distributed Approach to Motion Planning for Multirobot
Systems. IEEE Trans. Syst. Man Cybern. Syst. 2019,49, 2636–2660. [CrossRef]
30.
Umari, H.; Mukhopadhyay, S. Autonomous Robotic Exploration Based on Multiple Rapidly-exploring Randomized Trees.
In Proceedings of the IEEE/RJS International Conference on Intelligent Robots and Systems (IROS), Vancouver, BC, Canada,
24–28 September 2017.
