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Applied Mathematics and Computation

Hybrid genetic algorithm based on quantum computing for numerical optimization and parameter estimation

Ling Wang, Fang Tang, Hao Wu

Published Mar 31, 2005• 103 citations
Publisher
DOI
10.1016/j.amc.2005.01.115

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Quantum Genetic Algorithms (QGA) integrate genetic programming and quantum computing to address search and optimization problems. The standard strategy of the hybrid QGA approach is to add quantum resources to classical genetic algorithms (GA), thus improving their efficacy (i.e., quantum optimization of a classical algorithm). However, the extent of such improvements is still unclear. Conversely, Reduced Quantum Genetic Algorithm (RQGA) is a fully quantum algorithm that reduces the GA search for the best fitness in a population of potential solutions to running Grover's algorithm. Unfortunately, RQGA finds the best fitness value and its corresponding chromosome (i.e., the solution or one of the solutions of the problem) in exponential runtime, [[EQUATION]], where [[EQUATION]] is the number of qubits in the individuals' quantum register. This paper introduces a novel QGA optimization strategy, namely a classical optimization of a fully quantum algorithm, to address the Reduced Quantum Genetic Algorithm (RQGA) complexity problem. Accordingly, we control the complexity of theReduced Quantum Genetic Algorithm (RQGA) algorithm by selecting a limited number of qubits in the individuals' register and fixing the remaining ones as classical values of '0' and '1' with a genetic algorithm. We also improve the performance of RQGA by discarding unfit solutions and bounding the search only in the area of valid individuals. As a result, our Hybrid Quantum Algorithm with Genetic Optimization (HQAGO) solves search problems in [[EQUATION]] oracle queries, where [[EQUATION]] is the number of fixed classical bits in the individuals' register.
The genetic algorithm for numerical optimization (GA) of the metaheuristic class is a method for finding optimal solutions based on the biological principles of natural selection and variability. GA is characterized by high operating speed, resistance to noise in the data, low probability of hitting the local extremum of the multimodal objective function, as well as the simultaneous application of probabilistic and deterministic rules for generating search space points. An alternative to the classical GA is the quantum-inspired genetic algorithm for numerical optimization (QIGA), which has advantages that are unattainable for GA by using the concepts and principles of quantum computing. The article proposes a new approach to the implementation of a quantum-inspired genetic numerical optimization algorithm for searching for the global maximum of the objective function, based on modeling the functioning of the GA by simulating the execution of quantum calculations based on qudit in the conditions of the existence of quantum decoherence in the era of noisy medium-scale quantum algorithms. For this purpose, to carry out quantum operations of rotating the states of multilevel quantum systems, the paper presents a density matrix based on Heisenberg–Weyl operators as an analogue of the Bloch sphere for qudits. The simulation of quantum decoherence is interpreted from the point of view of the influence of extraneous noise emanating from the environment on the qudit and is presented as the use of a normal random variable with zero mathematical expectation and unit variance in quantum gates. At the same time, the work presents detailed pseudocodes of the functioning of both the most modified quantum-inspired genetic algorithm for numerical optimization and its individual operations. Testing is carried out by conducting computational experiments with the implementation of a modified algorithm on two-dimensional and multidimensional functions of test optimization problems, as well as when solving an applied optimization problem of planning hybrid flow production in the manufacturing industry based on financial costs and solving the problem of increasing forecasting accuracy based on compact extreme learning machines. The experimental results demonstrate the superiority of the new algorithm over QIGA and classical optimization algorithms in the accuracy of the solution, the speed of convergence with the target value of the global maximum and the execution time of the algorithm.