Keywords 

Quantum information; Fuzzy logic; Mobile robots. 

Introduction 

Following the publication of the classical work by Birkhoff and
von Neumann [1], the correspondence between the logic of quantum
mechanics and nonBoolean logics has attracted significant theoretical
and practical interest [2]. During the last decades, such an interest
was enhanced with the development of quantum information and
computation theory [3] and several approaches for the implementation
of quantum logic by the use of fuzzy methods were presented [48]. The
main goal of these studies is to construct a fuzzy model of quantum
mechanical operators such that it can be implemented by fuzzy
controllers [9]. In some cases, such models are complementary to
the implementation of fuzzy control models by the use of quantum
computations [10]. 

Recently, Hannachi et al. [5,6] suggested a direct fuzzy model of
quantum computations over the qubits with real amplitudes. In their
model, termed here as the H^{3}model, qubits are represented by pairs of
membership functions, and main quantum gates are implemented by
corresponding fuzzy operators on the unit square. Rybalov et al. [11]
supplement the H^{3}model with reverse fuzzy Hadamard operator and
implemented the resulting model in a mobile robot control system. The
resulting model is indicated henceforth as the extended H^{3}model. 

Although the extended H^{3}model provides basic representations
of quantum information concepts by fuzzy logic methods and, in the
completed form, can be used in certain applications, no general and
minimal twodimensional fuzzy model have been reported so far. 

The objective of the paper is to present a minimal fuzzy model on
a unit square that generalises the extended H^{3}model and, similar to
quantummechanical models, preserves smoothness of membership
functions and operators. The suggested model applies a parametric
system of the fuzzy not and phase operators, such that their compositions
provide a complete system of fuzzy operators and implement operators
of quantum information theory for real amplitudes. The fuzzy not and
phase operators are based on the uninorms [12]. 

The actions of the suggested minimal fuzzy model are illustrated
by the use of a real mobile robot control system. In addition, numerical
simulations and comparison with the control method based on the
extended H^{3}model [11] are provided. 

The obtained results can support a development of fuzzy logic
tools for the models of quantum information theory and their
implementation. 

The paper is organized as follows. The background includes a brief
presentation of the main fuzzy operators, which are included in the
H3model and extended H^{3}model. In Phase_{α} and not_{α} operators for
H3model, we defined the suggested parametric fuzzy not and phase
operators and consider their actions. Uninorm for Phase_{α} and Not_{α}
Operators and Projector section deals with the uninorm and the
metrics for the suggested operators. In implementation of the suggested
model and numerical analysis, an example, which illustrates an
implementation of the suggested model for the mobile robot’s control,
is presented. In addition, the section includes results of numerical
simulations and statistical comparison of the suggested model with the
extended H^{3}model applied for the mobile robot’s control. 

Background 

The states s of quantummechanical system [3] are defined by the
use of qubits that are two elements columnvectors 



where vectors and are the basic qubits and . The basic states and of the system are usually
denoted by (“spin up”) and (“spin down”): 



while the states and are defined as follows: 



The states form a complete system in such a
sense that each state can be obtained by the use of the other states [11],
i.e.: 





and, as it is usual in quantum mechanics, the states and are not distinguished. 

According to the Hannachi et al. [5,6], in the direct fuzzy model
of quantum computations to each quantum state with real
amplitudes fuzzy state , in which the membership functions and are defined as follows: 

, 

Inversely, given values of membership functions and , the
amplitudes and are obtained by the
following equations [5,6]: 





The Hannachi and the H^{3}model [5,6] include fuzzy representations
of the main qubit operators (Pauli operators I, X, Z, Hadamard H
operator and negative unitary operator I ) that are used in quantum
information theory [8]. The correspondence between quantum and
fuzzy operators is the following. Let us denote fuzzy operators by , , , and in correspondence to the quantum operators. Then
an application of the fuzzy operators to a fuzzy state results
in the following states [5,6]: 





Since the fuzzy state is not a vector, its row and column
representation are used here interchangeably. 

To complete the correspondence between quantum and fuzzy
models, Rybalov et al. [11] suggested the reverse fuzzy Hadamard
operator that corresponds to the quantum reverse Hadamard
operator. For a fuzzy state this operator is defined as
follows: 



The H^{3}model with the reverse Hadamard operator is called
extended H^{3}model. 

Phase_{α} and not_{α} operators for H^{3}model 

Both the original and the extended H^{3}models are rather complex.
Below we suggest a simpler fuzzy model that includes two operators, to
which the transformations that are provided by the extended H^{3}model
can be reduced. 

Let be a fuzzy state and assume that the values
of membership functions and are obtained according to the
H3model. However, the fuzzy states are not restricted
and the suggested operators can be applied for any pair of membership
functions . 

Let α be a real number such that 0 ≤α ≤ 1. For fuzzy state and the numberα, we define two operators phase_{α} and
not_{α} as follows: 





It is easy to verify that the operators phase_{α} and not_{α} have the
following properties: 





In addition, for the compositions of the operators phase_{α} and
not_{α} with any α ∈[0, 1], one obtains: 





In particular, for α = 0 and α = 1, the compositions of the
operators result in the following states: 





Let us apply the defined operators phase_{α} and not_{α} and their
compositions with α = 0 and α = 1 to the fuzzy states , , and , in which the values of membership functions are integer (including zero). The implementation of
the presented formulas results in the states with integer values of
membership functions, as it is shown in Figure 1. 

Now, let us present the actions of the operators phase_{α} and
not_{α} over the states , , and . For α = 0 andα = 1, these operators act as it is
shown in Figure 2. 

Finally, let us consider the actions of the operators phase_{α} and not_{α} with α = 1 2 . Direct calculations show that for the fuzzy states , , and with integer
values of membership functions the operators phase_{α=½} and not_{α=½}
result in the states that lie on the perimemter of the unit square. 

However, an application of the operators phase_{α=½} and not_{α=½}
to the resulting states , , and with noninteger values of membership functions
does not give the original states. The states that correspond to these
values are located in the internal part of the unit square instead of its
perimeter. To project such states to the perimeter let us define the
unique projector such that for any fuzzy state it results in
the state , which lies on the closest intersection
point of the perimeter, and a line that follows via the point and a center of the square (1 2 , 1 2). The strict meaning of the projector
is considered in the next section. The actions of the operators phase_{α} and not_{α} not with α= 1/2 (composed with the projector , i.e. under the
assumption that and are shown
in Figure 3. 

For convenience, the correspondence between quantum and H3
model operators and their representation by the use of phase_{α} and not_{α} operators is summarized in Table 1. 

The abovepresented consideration demonstrates that the
suggested phase_{α} and not_{α} operators completely implement the
transformations, which are provided by the extended H^{3}model that
includes fuzzy reverse Hadamard operator . 

Uninorm for Phasea and nota Operators and Projector 

Let us clarify the meaning of the aboveintroduced projecting
operator . Below we show that this operator provides a monotony property of the uninorms that are defined by compositions of the
operators phase_{α} and not_{α}. 

Let us start with the uninorm that is defined over the
membership functions. The uninorm R is defined as a mapping such that for any values of membership
functions , the following properties hold [12]: 

commutativity: 

monotonicity: 

associativity: 

existence of identity: for any μ ∈[0, 1] there exists ε ∈[0, 1] such
that R(μ,ε ) = μ. 

Let us define a similar uninorm U that is applicable to phase_{α} and not_{α} operators, which act on the pairs and result in the pairs . Below we denote these operators by
A, B, C and D. 

Assume that the uninorm U(A, B) is an operator over
the fuzzy states, which is defined as a composition of the
operators A, B, i.e. . Since, as indicated above,
it holds true that , and and acting of such uninorm
U on the phase_{α} and not_{α} not operators meets the commutativity and
associativity requirements, and an identity exists. 

The consideration of the monotonicity of the uninorm U requires
definition of the relations “<” and “>” between the operators. At first,
notice that the uninorm U for the operators acts on the pairs of pairs of
membership functions and results in the pair of membership functions,
i.e., 



In addition, notice, that the pair (1/2 , 1/2) represents a fixed point
for the operators phase_{α} and not_{α} not with any value of parameter α.
Hence, this point can be considered as a rapper or nullpoint. 

By the use of the point , the relations “<”
and “>” between the operators can be defined as follows. Let be a metric defined on the unit square, where , and . Then, for
any fuzzy state the distance from this state to the nullpoint is given by the formula: 



The relations “<” and “>” between the states and are specified by the use of the distances as follows. We
say that the state is greater than the state and write if in the defined metric d it holds true that 



and, similarly, we say that is less the state and write if
it holds true that 



If both and are satisfied, then we say that the
states are equal in the sense of distance from the nullpoint θ and write . 

The relations between the operators A and B that act over the fuzzy states,
and, correspondently, between the results of the applications of uninorm
U are defined by the relations between the resulting states as follows. Let and . Then, we say that A > B if and A < B if . 

Finally, let A, B, C and D be fuzzy operators, and let and Then, we say that 





Now, let us consider the role of the projecting operator . Assume
that the metric d is a Manhattan metric, i.e. 



Let be a state, and assume that fuzzy operators A, B, C
and D without projector act as follows: 



Then, by the use of the projector , , i.e., under an assumption that , one obtains: 



while the results of the operators C and D are still the same. Hence the
inequalities A > C and B > D hold both for original results and for
the states, which are obtained by the use of the projector . 

Direct calculations show that for the operators A, B, C and D
without projector it follows that 



Thus, for the operators without projector in spite of the
inequalities A > C and B > D, the relation U(A, B) >U(C, D) that is
required by monotonicity, in general, is not satisfied. 

However, similar considerations that include the projector result in the following inequalities m(U(A, B)) ≤ 1 andm(U(C, D)) ≤ 1. Notice that both and lie on the
perimeter of the unit square, i.e. they are boundary states. Moreover,
the results of the compositions and are also boundary states. Then, according to definitions of the operators
phase_{α} and not_{α}, the transformations of the states over the boundary
of the unit square are continuous. Hence, from the inequalities A > C
and B > D it follows thatU(A, B) >U(C, D), as it is required by the monotonicity property of the uninorm. This property of the suggested
operators allows their consideration in the framework of general fuzzy
operators. 

Implementation of the Suggested Model and Numerical
Analysis 

The suggested model that is based on phase_{α} and not_{α} operators,
as well as extended H3model, were implemented in a mobile robot
controller, and for both models comparative numerical simulations
and field experiments were carried out. As indicated above, the detailed
consideration of the experiments and simulation results is presented in
our work [11]. 

The correspondence between the robot’s states and its orientation
has been specified according to the quantum “spin orientations” as
follows: 

, , , . 

The implemented sequence of actions, in the terms of extended H^{3}
model, has the following form: 



According to this sequence, the robot conducts the following
actions: 

 moving one step forward; 

 turning to the right (direct Hadamard operator); 

 moving one step forward; 

 turning to the left (application of the direct Hadamard operator
over eastern orientation of the robot turns the robot to the left
returning it to the initial orientation; 

 moving one step forward; 

 turning to the right (in contrast to the direct Hadamard
operator, reverse Hadamard operator turns the robot left from
its initial orientation); 

 moving one step forward, 

and so on up to the last step when the robot returns to its initial state.
The obtained trajectory is simple enough for analysis, and, at the same
time, represents all possible turns of the robot, so that all transitions
between the states can be tested. 

In the field experiments, the mobile robot was programmed so that
it implemented the presented sequence of actions according to either
the extended H^{3}model or the suggested model based on phase_{α} and not_{α} operators. An example of the obtained trajectories of the robot,
while at the initial state it was oriented to the north, are shown in
Figure 4. 

The Figure demonstrates the difference between turns that are
governed by the extended H^{3}model and the suggested model based
on phase_{α} and not_{α} operators. Notice that the turns of the robot
controlled by phase_{α} and not_{α} model differ from the turns that were previously obtained for the model of quantum control [11], so the
topology of the states space for the phase_{α} and not_{α} not model differs
from both the extended H^{3} and the quantum models. 

Numerical analysis was focused on the errors of the final angle
that reflect the turns’ errors. The aboveindicated sequence of direct
and reverse Hadamard operators was implemented numerically, while
after each application of the operator a random error to the resulting
state was added. There were implemented three values of errors:
1% error, 5% error and 10% error. The 1% error corresponds to the
adding to the robot’s orientation a randomly chosen angle from the
interval (3.6°, 3.6°), the 5% error means adding a randomly chosen angle form the interval (18°, 18°), and 10% error corresponds to the
addition of the angle randomly chosen from the interval (36°, 36°). For
each interval, the simulation was trialed 1000 times and the differences
between the obtained and expected final orientations, i.e., the errors in
final orientation angles were logged. 

Histograms of the orientation errors in the final angle of the robots
are shown in Figure 5. The histograms in Figure 5a correspond to the
1% errors in the robot’s turns; the histograms in Figure 5b–to 5% turns’
errors; and the histograms in Figure 5c–to a 10% turns’ errors. 

By the use of the JarqueBera test with significance level 0.01, it
was found that in the case of 1% errors, the distributions of errors for
both models of control are approximately normal, while in cases of
5% and 10% errors the distributions of errors can not be considered as
normal. A series of ttests with a significance level of 0.01 showed that
the difference between the resulting errors is insignificant. Following
the KolmogorovSmirnov test with a significance level 0.01, in all three
cases of errors the obtained samples correspond to different continuous
distributions. 

The obtained results confirm our expectation that the model that is
based on phase_{α} and not_{α} operators acts differently in comparison
to the extended H^{3}model, while the results of application of the
suggested operators are equivalent. As indicated above, the difference
between the actions of the extended H^{3}model and the model based on
phase_{α} and not_{α} operators is provided by different topologies of the
states space. 

Conclusion 

In the paper, we suggested a system of fuzzy operators that
implements the operations over qubits. The system includes two
parameterized phase_{α} and not_{α} operators that act on the states
obtained by the extended H^{3}model. The transformations of the
states that correspond to the operators of the extended H3model
are determined by compositions of the defined phase_{α} and not_{α}
operators. For the suggested operators, the operational uninorm was
defined and analyzed, what introduces the suggested operators into
general fuzzy logic approach to quantum computations. 

The suggested system was illustrated by implementation for a mobile
robot control and was analyzed by numerical simulations. Comparison
of the control based on the suggested control demonstrated that the
results of the actions of the suggested minimal fuzzy system (that
includes two operators) are equivalent to the corresponding results
obtained by the extended H^{3}model (that includes six operators). 

The obtained results can support a development of fuzzy logic tools
for the models of quantum information theory. 

Acknowledgement 

We are thankful to Moshe Israel and Yaniv Reginiano for fruitful discussions.
The research was conducted in the CIM and Process Control Lab at Tel Aviv
University. 

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