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ISSN: 0974-7230
Journal of Computer Science & Systems Biology
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Fuzzy Implementation of Qubits Operators

Alexander Rybalov1, Eugene Kagan2, Alon Rapoport3 and Irad Ben-Gal4*

1Jerusalem College of Technology, Jerusalem, 91160, Israel

2Department of Industrial Engineering, Ariel University, Ariel, 40700, Israel

3Department of Mechanical Engineering, Tel-Aviv University, Ramat-Aviv, 69978, Israel

4Department of Industrial Engineering, Tel-Aviv University, Ramat-Aviv, 69978, Israel

*Corresponding Author:
Irad Ben-Gal
Department of Industrial Engineering
Aviv University, Ramat-Aviv, 69978, Israel
Tel: 13126136968
E-mail: [email protected]

Received Date: June 03, 2014; Accepted Date: June 25, 2014; Published Date: June 27, 2014

Citation: Rybalov A, Kagan E, Rapoport A, Ben-Gal I (2014) Fuzzy Implementation of Qubits Operators. J Comput Sci Syst Biol 7:163-168. doi: 10.4172/jcsb.1000151

Copyright: © 2014 Rybalov A, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In the paper, a complete minimal system of fuzzy logic operators that implements the operations over qubits is suggested. The model is based on a parametric system of the fuzzy not and phase operators and represents operators of quantum information theory acting on real amplitudes. An example of the system application is provided by its implementation to a mobile robot control, and its correspondence with the previously suggested models is illustrated by numerical simulations.

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 non-Boolean 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 [4-8]. 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 H3-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 H3-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 H3-model.

Although the extended H3-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 two-dimensional 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 H3-model and, similar to quantum-mechanical 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 H3-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 H3-model and extended H3-model. In Phaseα and notα operators for H3-model, we defined the suggested parametric fuzzy not and phase operators and consider their actions. Uninorm for Phaseα and Notα 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 H3-model applied for the mobile robot’s control.

Background

The states s of quantum-mechanical system [3] are defined by the use of qubits that are two elements column-vectors

Equation

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

Equation

while the states Equation and Equation are defined as follows:

Equation

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

Equation

Equation

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

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

Equation, Equation

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

Equation

Equation

The Hannachi and the H3-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 Equation, Equation, Equation, Equation and Equationin correspondence to the quantum operators. Then an application of the fuzzy operators to a fuzzy state Equation results in the following states [5,6]:

Equation

Equation

Since the fuzzy state Equation 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 Equationthat corresponds to the quantum reverse Hadamard operator. For a fuzzy state Equation this operator is defined as follows:

Equation

The H3-model with the reverse Hadamard operator Equation is called extended H3-model.

Phaseα and notα operators for H3-model

Both the original and the extended H3-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 H3-model can be reduced.

Let Equation be a fuzzy state and assume that the values of membership functions Equation and Equation are obtained according to the H3-model. However, the fuzzy states Equation are not restricted and the suggested operators can be applied for any pair of membership functions Equation.

Let α be a real number such that 0 ≤α ≤ 1. For fuzzy state Equation and the numberα, we define two operators phaseα and notα as follows:

Equation

Equation

It is easy to verify that the operators phaseα and notα have the following properties:

Equation

Equation

In addition, for the compositions of the operators phaseα and notα with any α ∈[0, 1], one obtains:

Equation

Equation

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

Equation

Equation

Let us apply the defined operators phaseα and notα and their compositions with α = 0 and α = 1 to the fuzzy states Equation, Equation, Equation and Equation, 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.

computer-science-systems-biology-membership-functions

Figure 1: Actions of phaseα and notα operators over the fuzzy states with integer values of membership functions.

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

computer-science-systems-biology-membership-functions

Figure 2: Actions of phaseα and notα operators over the fuzzy states with non-integer values of membership functions.

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

However, an application of the operators phaseα=½ and notα=½ to the resulting states Equation, Equation, Equation and Equation with non-integer 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 Equation such that for any fuzzy state Equation it results in the state Equation, which lies on the closest intersection point of the perimeter, and a line that follows via the point Equation 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 phaseα and notα not with α= 1/2 (composed with the projector Equation , i.e. under the assumption that Equation and Equation are shown in Figure 3.

computer-science-systems-biology-Actions-phase

Figure 3: Actions of phaseα and notα operators with α = 1 2 .

For convenience, the correspondence between quantum and H3- model operators and their representation by the use of phaseα and notα operators is summarized in Table 1.

Quantum operator Extended H3 model phase - not model
Equation Equation Equation or Equation,
α = 0 or α = 1
Equation Equation Equation,
α = 0 or α = 1
Equation Equation not0
Equation Equation not1
Equation Equation not1/2
Equation Equation Equation

Table 1: Correspondence between quantum, H3 and phaseα-notα models.

The above-presented consideration demonstrates that the suggested phaseα and notα operators completely implement the transformations, which are provided by the extended H3-model that includes fuzzy reverse Hadamard operator Equation.

Uninorm for Phasea and nota Operators and Projector Equation

Let us clarify the meaning of the above-introduced projecting operator Equation. Below we show that this operator provides a monotony property of the uninorms that are defined by compositions of the operators phaseα and notα.

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

commutativity: Equation

monotonicity: Equation

associativity: Equation

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 phaseα and notα operators, which act on the pairs Equation and result in the pairs Equation. 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. Equation. Since, as indicated above, it holds true that Equation, and Equation and acting of such uninorm U on the phaseα and notα not operators meets the commutativity and associativity requirements, and an identity Equation 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.,

Equation

In addition, notice, that the pair (1/2 , 1/2) represents a fixed point for the operators phaseα and notα not with any value of parameter α. Hence, this point can be considered as a rapper or null-point.

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

Equation

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

Equation

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

Equation

If both Equationand Equation are satisfied, then we say that the states are equal in the sense of distance from the null-point θ and write Equation.

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 Equation and Equation. Then, we say that A > B if Equation and A < B if Equation.

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

Equation

Equation

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

Equation

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

Equation

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

Equation

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 Equation.

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

Equation

Thus, for the operators without projector Equation 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 Equation result in the following inequalities m(U(A, B)) ≤ 1 andm(U(C, D)) ≤ 1. Notice that both Equation and Equation lie on the perimeter of the unit square, i.e. they are boundary states. Moreover, the results of the compositions Equation and Equation are also boundary states. Then, according to definitions of the operators phaseα and notα, 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 phaseα and notα operators, as well as extended H3-model, 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:

Equation, Equation, Equation, Equation.

The implemented sequence of actions, in the terms of extended H3- model, has the following form:

Equation

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 H3-model or the suggested model based on phaseα and notα 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.

computer-science-systems-biology-robot-movements

Figure 4: Trajectories of the robot movements controlled by the extended H3-model and by the model based on of phaseα and notα operators. White arrow denotes a starting position and orientation of the robot, and a black arrow denotes its final position and orientation.

The Figure demonstrates the difference between turns that are governed by the extended H3-model and the suggested model based on phaseα and notα operators. Notice that the turns of the robot controlled by phaseα and notα 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 phaseα and notα not model differs from both the extended H3 and the quantum models.

Numerical analysis was focused on the errors of the final angle that reflect the turns’ errors. The above-indicated 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.

computer-science-systems-biology-final-angle

Figure 5: Histograms of the final angle errors with different percentage of turn’s errors.

By the use of the Jarque-Bera 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 t-tests with a significance level of 0.01 showed that the difference between the resulting errors is insignificant. Following the Kolmogorov-Smirnov 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 phaseα and notα operators acts differently in comparison to the extended H3-model, while the results of application of the suggested operators are equivalent. As indicated above, the difference between the actions of the extended H3-model and the model based on phaseα and notα 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 phaseα and notα operators that act on the states obtained by the extended H3-model. The transformations of the states that correspond to the operators of the extended H3-model are determined by compositions of the defined phaseα and notα 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 H3-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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