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Electronic Notes in Theoretical Computer Science

Interpretations on Quantum Fuzzy Computing: Intuitionistic Fuzzy Operations × Quantum Operators

Renata Reiser, Alexandre Lemke, Anderson Ávila, Júlia Krüger Vieira, Mauricio Lima Pilla, André Rauber Du Bois

Published Sep 1, 2016• 11 citations
Publisher

Abstract

Quantum processes provide a parallel model for fuzzy connectives. Calculations of quantum states may be simultaneously performed by the superposition of membership and non-membership degrees of each element regarding the intuitionistic fuzzy sets. This work aims to interpret Atanassov's intuitionistic fuzzy logic through quantum computing, where not only intuitionistic fuzzy sets, but also their basic operations and corresponding connectives (negation, conjuntion, disjuntion, difference, codifference, implication, and coimplication), are interpreted based on the traditional quantum circuit model.

DOI
10.1016/j.entcs.2016.09.012

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Quantum processes provide a parallel model for fuzzy connectives. Calculations of quantum states may be simultaneously performed by the superposition of membership and non-membership degrees of each element regarding the intuitionistic fuzzy sets. This work aims to interpret Atanassov's intuitionistic fuzzy logic through quantum computing, where not only intuitionistic fuzzy sets, but also their basic operations and corresponding connectives (negation, conjuntion, disjuntion, difference, codifference, implication, and coimplication), are interpreted based on the traditional quantum circuit model.
This paper shows that quantum computing extends the class of fuzzy sets, taking advantage of properties such as quantum parallelism. The central idea associates the states of a quantum register with membership functions of fuzzy subsets, and the rules for the processes of fuzzyfication are performed by unitary quantum transformations. Besides studying the construction of logical operators, such as negation, using quantum, this paper also introduces the definition of t-norms and t-conorms based on unitary and controlled quantum gates. Such constructors allow modelling and interpreting union, intersection and difference between fuzzy sets. As the main contribution, an interpretation for the Reichenbach implication from quantum computing is obtained. An evaluation of the corresponding computation is implemented and simulated in the VPE-qGM visual programming environment.
Computer systems based on fuzzy logic are capable of generating a reliable output even when handling inaccurate input data by applying a rule based system. The main contribution of this paper is to show that quantum computing can be used to extend the class of fuzzy sets. The central idea associates the states of a quantum register to membership functions (mFs) of fuzzy subsets, and the rules for the processes of fuzzyfication are performed by unitary quantum transformations. Thus, this paper describes multi-dimensional quantum registers, associated to mFs on the unitary interval U, in order to introduce a novel interpretation of aggregations found in fuzzy set theory. In particular, t-norms and t-conorms based on quantum gates allows the modeling and interpretation of union, intersection, difference and implication among fuzzy sets, also including an expression for the class of fuzzy S-implications. Furthermore, an interpretation of the symmetric sum was achieved by considering the quantum register sum operator.
In this paper, the formulation of Quantum Mechanics in terms of fuzzy logic and fuzzy sets is explored. A result by Pykacz, which establishes a correspondence between (quantum) logics (lattices with certain properties) and certain families of fuzzy sets, is applied to the Birkhoff–von Neumann logic, the lattice of projectors of a Hilbert space. Three cases are considered: the qubit, two qubits entangled, and a qutrit ‘nested’ inside the two entangled qubits. The membership functions of the fuzzy sets are explicitly computed and all the connectives of the fuzzy sets are interpreted as operations with these particular membership functions. In this way, a complete picture of the standard quantum logic in terms of fuzzy sets is obtained for the systems considered.
In this paper we present our proposal for a fuzzy logic model based on Quantum Computing. After analyzing how logical connectives are constructed and applied in fuzzy logic, we research how Quantum Computing can provide a new model to manage fuzzy variables, mainly using superposition of states for the implementation of fuzzy information. A comparison between our proposal and several classical connectives is carried out, including negation, conjunction and disjunction. We prove that quantum fuzzy connectives present a proper behavior, manifesting the peculiarities of the quantum paradigm, which is beneficial for the representation of uncertain knowledge. We conclude on how this work establishes a model that can be further expanded for the development of quantum fuzzy reasoning.
This study is given at the intersection of three important areas: Modal Logic (ML), Fuzzy Logic (FL), and Quantum Computing (QC), leveraging from their main features. On the one hand, we have the QC ability to handle complex data more efficiently, taking advantage of the quantum mechanics concepts. On the other hand, ML and FL allow us to express uncertainties by mathematically modeling the imprecision of the natural language. Therefore, we first provide an algebraic model to interpret modal operators on fuzzy logic, and then we represent them in a quantum computing environment. Besides, we present some case studies simulating the fuzzy modal connectives in the Qiskit platform, aiming to better understand the evolution of quantum circuits.
An approach for emulating quantum circuits using conventional analog hardware is presented based on the intuitive similarity between fuzzy logic and quantum superposition, as well as some geometrical analogies. This approach has the advantage of being easy to implement on dedicated hardware for parallel processing of membership functions and fuzzy inference, compared to conventional quantum computing, which requires quantum mechanical systems which are extremely sensitive to noise and difficult to extend to large scale systems. Using geometrical analogies and a suitable transformation, qubits are modeled as pairs of fuzzy membership functions evolving on the unit square and basic one qubit gates are modeled as transformations on this unit square. A fuzzy implementation of the one bit and two bit Deutsch-Jozsa algorithm is proposed. Physical implementation and advantages, such as the possibility of implementing nonlinear or non-unitary gates, as well as drawbacks of the proposed model compared to conventional quantum computing are shown.