The goal of this thesis is to present evolutionary algorithms, and demonstrate their applicability in quantum computing.As an introduction to evolutionary algorithms, it is applied to the simple but still challenging (from a computational viewpoint) Travelling Salesman Problem (TSP).This example is used to illustrate the effect of various parameters like selection method, and maximum population size on the accuracy and efficiency of the evolutionary algorithms.For the sample problem, the 48 continental state capitals of the USA, solutions are evolved and compared to the known optimal solution.From this investigation tournament selection was shown to be the most effective selection method, and that a population of 200 individuals per generation gave the most effective convergence rates.In the next part of the thesis, evolutionary algorithms are applied to the generation of optimal quantum circuits for the following cases:• The identity transformation : Picked for its simplicity as a test of the correct implementation of the evolutionary algorithm.The results of this investigation showed that the solver program functions correctly and that evolutionary algorithms can indeed find valid solutions for this kind of problem.• The work by Ding et al. [16] on optimal circuits for the two-qubit entanglement gate, controlled-S gate as well as the three qubit entanglement gate are solved by means of EA and the results compared.In all cases similar circuits are produced in fewer generations than the application of Ding et al. [16].The three qubit quantum Fourier transform gate was also attempted, but no convergence was attained.• The quantum teleportation algorithm is also investigated.Firstly the nature of the transformation that leads to quantum teleportation is considered.Next an effective circuit is sought using evolutionary algorithms.The best result is one gate longer than Brassard [11], and seven gates longer than Yabuki [61].iii
The goal of this thesis is to present evolutionary algorithms, and demonstrate their applicability in quantum computing.As an introduction to evolutionary algorithms, it is applied to the simple but still challenging (from a computational viewpoint) Travelling Salesman Problem (TSP).This example is used to illustrate the effect of various parameters like selection method, and maximum population size on the accuracy and efficiency of the evolutionary algorithms.For the sample problem, the 48 continental state capitals of the USA, solutions are evolved and compared to the known optimal solution.From this investigation tournament selection was shown to be the most effective selection method, and that a population of 200 individuals per generation gave the most effective convergence rates.In the next part of the thesis, evolutionary algorithms are applied to the generation of optimal quantum circuits for the following cases:• The identity transformation : Picked for its simplicity as a test of the correct implementation of the evolutionary algorithm.The results of this investigation showed that the solver program functions correctly and that evolutionary algorithms can indeed find valid solutions for this kind of problem.• The work by Ding et al. [16] on optimal circuits for the two-qubit entanglement gate, controlled-S gate as well as the three qubit entanglement gate are solved by means of EA and the results compared.In all cases similar circuits are produced in fewer generations than the application of Ding et al. [16].The three qubit quantum Fourier transform gate was also attempted, but no convergence was attained.• The quantum teleportation algorithm is also investigated.Firstly the nature of the transformation that leads to quantum teleportation is considered.Next an effective circuit is sought using evolutionary algorithms.The best result is one gate longer than Brassard [11], and seven gates longer than Yabuki [61].iii
This paper implements a new way of solving a problem called the traveling salesman problem (TSP) using quantum genetic algorithm (QGA). We compared how well this new approach works to the traditional method known as a classical genetic algorithm (CGA). The TSP is a well-established challenge in combinatorial optimization where the objective is to find the most efficient path to visit a series of cities, minimizing the total distance, and returning to the starting point. We chose the TSP to test the performance of both algorithms because of its computational complexity and importance in practical applications. We choose the dataset from the international standard library TSPLIB for our experiments. By designing and implementing both algorithms and conducting experiments on various sizes and types of TSP instances, we provide an in-depth analysis of the accuracy of the optimal solution, the number of iterations, the execution time, and the stability of the algorithms for both. The empirical findings indicate that the CGA outperforms the QGA in terms of finding superior solutions more quickly in most of the test instances, especially when the problem size is large. This suggests that although the principle of quantum computing provides a new way to solve complex combinatorial optimisation problems, the implementation of quantum phenomena and the setting of parameters such as the optimal angle for a quantum revolving gate is challenging and need further optimisation to achieve the desired results. Additionally, it is important to note that the QGA has not been tested on real quantum hardware, so its true performance remains unverified. These limitations provide rich opportunities for further research in the future.
Quantum-Inspired Evolutionary Algorithms (QIEAs) represent a novel fusion of quantum computing principles and classical evolutionary strategies, offering a powerful approach to solving complex optimization problems. This chapter provides a comprehensive overview, detailing foundational concepts, implementation techniques, and potential applications of QIEAs. By leveraging quantum phenomena such as superposition and entanglement, QIEAs enhance the exploitation and exploration capabilities of traditional evolutionary algorithms (EAs), thus enabling an efficient search of vast solution spaces. The study examines key components of QIEA, including quantum encoding, rotation gates, and entanglement operations, and compares its performance with classical EAs. By analyzing real-life examples of combinatorial optimization problems (COPs) such as the minimum spanning tree (MST), traveling salesman problem (TSP), and graph coloring, the chapter demonstrates QIEA’s superior efficiency and solution quality. Despite implementation challenges, QIEA shows promise in various domains, including machine learning, finance, healthcare, telecommunications, and e-commerce. Future research directions are outlined, emphasizing the need for simplified quantum-inspired operators, hybrid quantum-classical systems, and interdisciplinary collaboration. This chapter highlights the transformative potential of QIEA, setting the stage for further advancements and applications in optimization and computational efficiency.
The goal of this work is to understand the application of the evolutionary programming approach to the problem of quantum circuit design. This problem is motivated by the following observations: In order to keep up with the seemingly insatiable demand for computing power our computing devices will continue to shrink, all the way down to the atomic scale, at which point they become quantum mechanical systems. In fact, this event, known as Moore?s Horizon, is likely to occur in less than 25 years. The recent discovery of several quantum algorithms which can solve some interesting problems more efficiently than any known classical algorithm. While we are not yet certain that quantum computers will ever be practical to build, there do now exist the first few astonishing experimental devices capable of briefly manipulating small quantities of quantum information. The programming of these devices is already a nontrivial problem, and as these devices and their algorithms become more complicated this problem will quickly become a significant challenge. The Evolutionary Programming (EP) approach to problem solving seeks to mimic the processes of evolutionary biology which have resulted in the awesome complexity of living systems, almost all of which are well beyond our current analysis and engineering capabilities. This approach is motivated by the highly successful application of Koza?s Genetic Programming (GP) approach to a variety of circuit design problems, and specifically the preliminary reports byWilliams and Gray and also Rubinstein who applied GP to quantum circuit design. Accompanying this work is software for evolutionary quantum circuit design which incorporates several advances over previous approaches, including: A formal language for describing parallel quantum circuits out of an arbitary elementary gate set, including gates with one or more parameters. A fitness assessment procedure that measures both average case fidelity with a respect for global phase equivalences, and implementation cost. A Memetic Programming (MP) based reproductive strategy that uses a combination of global genetic and local memetic searches to effectively search through diverse circuit topologies and optimize the parameterized gates they contain. Several benchmark experiments are performed on small problems which support the conclusion that Evolutionary Programming is a viable approach to quantum circuit design and that further experiments utilizing more computational resources and more problem insight can be expected to yield many new and interesting quantum circuits.
Automatic quantum circuit design is motivated by the difficulties in manual design, because quantum algorithms are highly non-intuitive and practical quantum computer hardware is not yet available. Thus, quantum computers have to be simulated on classical hardware which naturally entails an exponential growth of computational costs and allows only to simulate small quantum systems, i. e., with only few qubits. Huge search spaces render evolutionary approaches nearly unable to achieve breakthrough solutions in the development of new quantum algorithms. Consequently, at present we must be content to evolve essentially already existing (black-box) quantum algorithms. This thesis presents empirical results on the evolution of quantum circuits using genetic programming. For that purpose, a linear and a linear-tree GP system (allowing intermediate measurements) with integrated quantum computer simulator were implemented. Their practicality in evolving quantum circuits is shown in different experiments for 1-SAT (solutions act like Hogg's algorithm) and the Deutsch-Jozsa problem. These experiments confirm that the evolution of quantum circuits is practically feasible only for sufficiently small problem instances. In this context, scalability and the detection of scalability becomes very important. It is shown that scalable quantum circuits are evolvable to a certain degree: a general quantum circuit can be inferred manually from the evolved solutions for small instances of the given problem. Besides, further experiments indicate that 're-evolution' is effective for the evolution of scalable quantum circuits. With this method the start population of a problem instance is inoculated with evolved solutions for a smaller problem instance. Furthermore, investigations of fitness landscapes and selection strategies are made, with the aim of improving the efficiency of evolutionary search. A notable result is that using the crossover operator damages rather than benefits evolution of quantum circuits.
Simulating quantum mechanical evolutions in general is difficult on classical computers because the size of the Hilbert space grows exponentially with the number of particles involved in the simulations. However, a quantum computer, which obeys the laws of quantum mechanics, is believed to perform such simulations more efficiently. The time evolution operator, a unitary matrix, of a quantum system acts as a quantum gate changing the state of the system in time. Therefore, any computation on quantum computers can be interpreted as a unitary transformation on quantum state vectors. Generally, quantum gates are sufficiently simple operations so as to avoid difficulty in the physical implementation. The implementation of a general given computation represented by a unitary matrix on quantum computers requires finding an array of elementary quantum gates describing the desired computation. This describes a matrix decomposition problem known as the quantum circuit design problem. In this dissertation, we address the following questions concerning quantum circuit designs: How hard is it to find suitable quantum circuits for a given computation? What are the possible circuit design methods? How do we apply these to simulations of quantum problems, in particular quantum chemistry problems? To answer these questions, we categorize methods for designing quantum circuits as evolutionary algorithms and deterministic methods. We apply our developed group leaders optimization algorithm, an evolutionary algorithm, to find quantum circuits for known quantum algorithms and the simulation of molecular Hamiltonians. Then, we present a deterministic circuit design approach to produce universal circuits, which can be programmable. Moreover, we show how to use these programmable circuit designs within the quantum phase estimation algorithm to find eigenvalues of nonunitary matrices and study resonances in quantum systems. Finally, the quantum phase estimation algorithm is employed to efficiently solve ranking problems: in particular, multiple network alignment is considered to find the similarities between protein-protein interaction networks and molecular networks.
Optimization is one of the most primary jobs of all the engineering problems. We are fortunate to have blessed with scores of methods available for optimization purpose. Evolutionary algorithm is fast becoming one of the most sought after methods for this purpose. This method depends upon random moves. So sometimes the time consumed by this algorithm is much more than expected. With an aim to get rid of this problem, in this paper we propose an entirely new approach of evolutionary algorithm based on quantum computing. A quantum evolutionary algorithm can be termed as a concept which utilizes certain tools of quantum computing for search and optimization purposes. A quantum algorithm relies heavily on qubits and quantum gates. Qubits are the entities which actually contain information in quantum computing. They can either remain in a secluded state or a superposition of two or more qubits states. When they are in a superposition state we employ the concepts of probability to predict where they can actually be found. On the other hand quantum gates utilize qubits to achieve some predefined goals. These predefined goals in case of an optimization problem are to maximize or minimize some sort of objective function. This paper proposes a novel evolutionary algorithm employing quantum computing to address optimization problems. We have tested our algorithm on a famous problem known as Gear Train Design Problem. The results so obtained outperform the classical optimization methods.