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IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)

Implementing Defuzzification Operators on Quantum Annealers

Amir Pourabdollah, Giovanni Acampora, Roberto Schiattarella

Published Jul 18, 2022• 10 citations
Publisher

Abstract

Due to the built-in parallelism of quantum computing, there is an unexplored potential for some complex fuzzy logic computations to take the advantage of the future quantum computers. Recently, it has been introduced a novel representation of fuzzy sets and implementations of some basic fuzzy logic operators (union, intersection, alpha-cut and maximum) based on solving a Quadratic Unconstrained Binary Optimization (QUBO) problems, on a type of quantum computers known as quantum annealers. In this paper, this work is extended by presenting an implementation of centroid defuzzification on quantum annealer machines, based on binary quadratic model (BQM) but this time using Ising model. Having the basic operations and defuzzification implemented on quantum computers, this paper paves the way towards the implementation of a whole fuzzy inference engine on enhanced devices, such as quantum annealers.

DOI
10.1109/fuzz-ieee55066.2022.9882576

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Due to the built-in parallelism of quantum computing, there is an unexplored potential for some complex fuzzy logic computations to take the advantage of the future quantum computers. Recently, it has been introduced a novel representation of fuzzy sets and implementations of some basic fuzzy logic operators (union, intersection, alpha-cut and maximum) based on solving a Quadratic Unconstrained Binary Optimization (QUBO) problems, on a type of quantum computers known as quantum annealers. In this paper, this work is extended by presenting an implementation of centroid defuzzification on quantum annealer machines, based on binary quadratic model (BQM) but this time using Ising model. Having the basic operations and defuzzification implemented on quantum computers, this paper paves the way towards the implementation of a whole fuzzy inference engine on enhanced devices, such as quantum annealers.
Quantum computers can potentially perform certain types of optimisation problems much more efficiently than classical computers, making them a promising tool for solving complex fuzzy logic problems. In two recent developments, based on solving Quadratic Unconstrained Binary Optimization (QUBO) problems on a type of quantum computers known as quantum annealers, we have introduced novel representations of a) fuzzy sets; b) implementations of some basic fuzzy logic operators (union, intersection, alpha-cut and maximum) and; c) the centroid defuzzification. In this paper, the previous works are further extended by presenting an implementation of Mamdani inference on the quantum annealer machines. We first present how the fuzzy rules can be formulated for such an implementation, then we present how to cascade different quantum-fuzzy operators in order to implement the quantum-fuzzy inference, and finally, a sample implementation of the inference on a real quantum computer is demonstrated. Having the main components of a rule-based fuzzy logic system implemented on quantum computers, this paper provides an integrated solution for implementing a whole fuzzy rule-based system on quantum computers.
Quantum computation is going to revolutionize the world of computing by enabling the design of massive parallel algorithms that solve hard problems in an efficient way, thanks to the exploitation of quantum mechanics effects, such as superposition, entanglement, and interference. These computational improvements could strongly influence the way how fuzzy systems are designed and used in contexts, such as Big Data, where computational efficiency represents a nonnegligible constraint to be taken into account. In order to pave the way toward this innovative scenario, this article introduces a novel representation of fuzzy sets and operators based on quadratic unconstrained binary optimization problems, so as to enable the implementation of fuzzy inference engines on a type of quantum computers known as quantum annealers.
Karnik-Mendel method (KM) is widely known for type-reduction of interval type-2 fuzzy sets. The original and enhanced solutions of KM use various forms of iteration to converge onto the result, leading to different levels of computational complexities. Based on the algorithmic advancements in quantum computing, this paper proposes a non-iterative quantum computing solution to KM. We map the KM problem to the problem of minimising an objective function in the form of a binary constrained quadratic model (CQM) which can then be solved in a single run using “quantum annealing”. The algorithms used in each step, together with a numerical example and the results from real quantum computer are provided. Quantum technology permitting in the future, the proposed solution can potentially remove the known defuzzification bottleneck of designing type-2 fuzzy systems.
Most quantum annealers, or Ising machines have the hardware limitations in its qubit size and thus the computable problem size is much limited. This paper proposes a novel classical-Ising hybrid annealing method which virtually extends the computable size of Ising machines based on a theoretical analysis. Given a quadratic unconstrained binary optimization (QUBO) model, the proposed method selects the cutting binary variable that is most likely to have the same value as the ground-state solution. Then, we recursively cut the QUBO model based on the cutting binary variable until we obtain sufficiently small-sized subQUBO models. Experimental evaluations demonstrate that the proposed hybrid annealing method can give much better quasi-ground-state solutions than state-of-the-art existing methods for large-sized QUBO models.
Quantum annealing is a method based on simulated annealing where temperature variations are replaced by quantum fluctuations that cause qubit state transitions. ISING problems and Quadratic Unconstrained Binary Optimization problems (QUBO) can be tackled by quantum annealing-based machines. We can link the interconnected qubits of the machine and the binaries of our model: qubits are binary variables and each pair of them linked by a coupler has a strong impact on the equality or the non-equality of the two binaries of the associated pair. Quantum annealing is based on the fact that any system tends to seek its minimum energy state. Starting from qubits in a state of superposition where all the solutions to the problem are fairly represented, the machine will apply a magnetic field by targeting the qubits and couplers in such a way as to make their value energetically favorable in the direction of optimization (minimization). For the coupled qubits, it is the quadratic products of the binary variables that are considered here and the physical system will make it energetically favorable for them to take (or not) the same values. To work on these machines based on quantum annealing in order to solve optimization problems with constraints, the latter must be relaxed while penalizing the Objective function. After presenting the case of the Max-Cut problem, we will look at such constrained problems.
Recent advancements in quantum computing have demonstrated significant potential for solving combinatorial optimization problems, like the quadratic knapsack problem, a constrained binary optimization problem. How- ever, current quantum and quantum-inspired algorithms often require transforming these constrained problems into an unconstrained form, known as Quadratic Unconstrained Binary Optimization (QUBO). Such transfor- mations can significantly impact the algorithms’ speed and efficiency. In this study, we evaluate five existing transformation methods and propose four novel approaches. We assess all nine methods using Simulated An- nealing and find that three of our approaches outperform existing methods in terms of execution time and the quality and quantity of feasible solutions found. Additionally, we tested these transformations on quantum annealers, which were unable to solve even small problem instances, due to limitations in connectivity and error rates. However, our results highlight the advantages of the new approaches, which reduce the total num- ber of variables in the QUBO representation. This is a critical factor for enhanced performance on emerging quantum hardware, since it also reduces the required number of qubits and the embedding chain lengths.