Fuzzy Implementation of Qubits Operators

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].


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][5][6][7][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 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 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 H 3 -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 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 H 3 -model and extended H 3 -model. In Phase α and not α operators for H 3 -model, 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 quantum-mechanical system [3] are defined by the use of qubits that are two elements column-vectors The basic states 0 and 1 of the system are usually denoted by " " ↑ ("spin up") and " " ↓ ("spin down"): while the states " " ← and " " → are defined as follows: 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 Ĩ , Since the fuzzy state s 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]

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.
It is easy to verify that the operators α phase and α not have the following properties: In particular, for 0 = α and 1 = α , the compositions of the operators result in the following states:    . Direct calculations show that for the fuzzy states ( ) . Below we denote these operators by A, B, C and D.
Assume that the uninorm 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 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 ( ) 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 . Then, we say that Finally, let A, B, C and D be fuzzy operators, and let Then, we say that Now, let us consider the role of the projecting operator P . Assume that the metric d is a Manhattan metric, i.e. Then, by the use of the projector P , i.e., under an assumption that , one obtains: Thus, for the operators without projector P in spite of the inequalities C A > and D B > , the relation ( ) ( ) that is required by monotonicity, in general, is not satisfied.
However, similar considerations that include the projector P result in the following inequalities Notice that both ( ) ( ) , 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 α phase and α not operators, as well as extended H 3 -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: north, The implemented sequence of actions, in the terms of extended H 3model, 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 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 ( ) , the 5% error means adding a randomly chosen   . 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 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

Conclusion
In the paper, we suggested a system of fuzzy operators that implements the operations over qubits. The system includes two parameterized 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.