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Fuzzy Computing from Quantum Computing - Case Study in Reichenbach Implication Class

Adriano Maron, Lidiane Visintin, Renata Reiser, Ana Maria Barbosa Abeijon

Published Jan 1, 2013• 4 citations
OpenAlex

Abstract

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.

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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.
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.
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.
In this article we investigate a way in which quantum computing can be used to extend the class of fuzzy sets. The core idea is to see states of a quantum register as characteristic functions of quantum fuzzy subsets of a given set. As the real unit interval is embedded in the Bloch sphere, every fuzzy set is automatically a quantum fuzzy set. However, a generic quantum fuzzy set can be seen as a (possibly entangled) superposition of many fuzzy sets at once, offering new opportunities for modeling uncertainty. After introducing the main framework of quantum fuzzy set theory, we analyze the standard operations of fuzzification and defuzzification from our viewpoint. We conclude this preliminary paper with a list of possible applications of quantum fuzzy sets to pattern recognition, as well as future directions of pure research in quantum fuzzy set theory.
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.
Quantum computers can be a revolutionary tool to implement inference engines for fuzzy rule-based systems. In fact, the use of quantum mechanical principles can enable parallel execution of fuzzy rules and allow them to be used efficiently in complex contexts such as distributed and big data environments. This article introduces the very first quantum-based fuzzy inference engine that is capable of providing exponential acceleration in fuzzy rule execution compared to its classical counterpart, and allows a quantum computer to be programmed by fuzzy linguistic rules. The proposed inference engine was implemented using a quantum algorithm design scheme based on the oracle notion. This scheme allows the modeling of a fuzzy rule-based system as a Boolean function, the oracle, which is able to reconstruct the relationships between the antecedent and consequent parts of fuzzy rules, and can be efficiently computed on a quantum computer. The suitability of the proposed quantum algorithm for use as a fuzzy inference engine was tested in a typical benchmark scenario, such as that provided by inverted pendulum control.