Genetic algorithms (GA) are computational methods for solving optimization problems inspired by natural selection. Because we can simulate the quantum circuits that implement GA in different highly configurable noise models and even run GA on actual quantum computers, we can analyze this class of heuristic methods in the quantum context for NP-hard problems. This paper proposes an instantiation of the Reduced Quantum Genetic Algorithm (RQGA) that solves the NP-hard graph coloring problem in O(N 1/2 ). The proposed implementation solves both vertex and edge coloring and can also determine the chromatic number ( i.e ., the minimum number of colors required to color the graph). We examine the results, analyze the algorithm convergence, and measure the algorithm's performance using the Qiskit simulation environment. Our Reduced Quantum Genetic Algorithm (RQGA) circuit implementation and the graph coloring results show that quantum heuristics can tackle complex computational problems more efficiently than their conventional counterparts.
Genetic algorithms (GA) are computational methods for solving optimization problems inspired by natural selection. Because we can simulate the quantum circuits that implement GA in different highly configurable noise models and even run GA on actual quantum computers, we can analyze this class of heuristic methods in the quantum context for NP-hard problems. This paper proposes an instantiation of the Reduced Quantum Genetic Algorithm (RQGA) that solves the NP-hard graph coloring problem in O(N 1/2 ). The proposed implementation solves both vertex and edge coloring and can also determine the chromatic number ( i.e ., the minimum number of colors required to color the graph). We examine the results, analyze the algorithm convergence, and measure the algorithm's performance using the Qiskit simulation environment. Our Reduced Quantum Genetic Algorithm (RQGA) circuit implementation and the graph coloring results show that quantum heuristics can tackle complex computational problems more efficiently than their conventional counterparts.
Genetic Algorithm (GA) are common probabilistic optimization methods inspired by the process of natural selection. Quantum computers promise substantial speedups over conventional machines, and libraries allow the emulation of circuits on a quantum computer in different highly configurable noise models and even run on quantum computers. Therefore, we need to analyze this class of heuristic methods in the quantum context. We propose a circuit level implementation of the Reduced Quantum Genetic Algorithm (RQGA) using Python and Qiskit. Our main goal is to analyze the circuit complexity from the perspectives of the number of qubits required and the number of quantum gates used. To achieve our goal, we instantiate the framework for solving the knapsack problem, examine the results in a simulated environment, and analyze the circuit's complexity.
Abstract Quantum Genetic Algorithms (QGAs) integrate genetic programming and quantum computing to address search and optimization problems. Most QGA approaches add quantum features to genetic algorithm operators, such as selection, crossover, or mutation. Oppositely, the Reduced Quantum Genetic Algorithm (RQGA) is a fully quantum algorithm that encodes the entire search space (i.e., population) as a superposition of all possible solutions (chromosomes) using the individuals’ quantum register; the fitness function takes the individuals’ quantum register as input to generate corresponding fitness values within the chromosomes’ superposition. RQGA finds the best fitness value and its corresponding chromosome (i.e., the solution or one of the solutions) using Grover’s algorithm. The complexity of Grover’s algorithm is O(2N⁄2), where N is the number of superposed individual chromosomes. However, RQGA operates on an exponentially-big search space (N = 2n, where n is the number of qubits in the individuals’ quantum register), which entails an exponential runtime of O(2n⁄2). This paper introduces an optimization solution for RQGA that controls the algorithm complexity 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 the RQGA by discarding unfit solutions and bounding the search only in the valid individuals’ area. Putting it all together, we introduce a novel quantum genetic algorithm—Hybrid Quantum Algorithm with Genetic Optimization (HQAGO)—that solves search problems in O(2(n-k)⁄2) oracle queries, where k is the number of fixed classical bits in the individuals’ register. We illustrate instantiations of HQAGO that solve the NP-hard knapsack and graph coloring problems, analyze the complexity of the new algorithm, and study the convergence of its heuristic part.
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.
Abstract Graph coloring with minimum colors is a well-known NP-complete problem in computer science and mathematics, having diverse real-life applications such as scheduling, register allocation, and frequency assignment. However, a classical computer requires exponential time to solve the graph coloring problem with the minimum number of colors. Therefore, classical approaches become infeasible for solving this problem on large graphs. In contrast, quantum computing offers the potential for speedup on specific problems, although solutions are typically obtained with a certain probability of correctness. This work proposes a quantum–classical method for solving the graph coloring problem and finding the chromatic number of a graph with guaranteed correctness, and also reduces the time and qubit complexities. The proposed method is inspired by Randomized Grover’s search algorithm, and the classical binary search algorithm. Here, Randomized Grover’s search algorithm provides quadratic speedup and this algorithm is useful in real situations when the number of solutions is not known in advance in contrast to the Grover’s search algorithm. The binary search algorithm helps to achieve logarithmic speedup, which reduces overall complexity. The experimental results strongly validate our claims and demonstrate the superiority of the approach. Furthermore, a comprehensive mathematical analysis is carried out to examine its complexities, as well as to compare its computational efficiency with existing methods.
The development in the field of quantum computing gives us a significant edge over classical computing in terms of time and efficiency.This is particularly useful for NP-hard problems such as graph layout problems.Since many real world problems are effectively solved by genetic algorithm (GA) and the performance of GA highly depends upon the setting of its parameters, therefore this paper focuses on a Quantum Inspired Genetic Algorithm (QIGA) and develops and evaluates adaptive strategies for the same.QIGA adapts ideas of Q-bits, superposition of Q-bits from quantum computing.The effectiveness and the applicability of adaptive QIGA is demonstrated by experimental results on the benchmark Knapsack, Maxcut and Onemax combinatorial optimization problems.The results show that adaptive QIGA is superior to QIGAs.
This paper presents a comprehensive review of recent advancements in Genetic Quantum Algorithms (GQAs) and their application to combinatorial optimization problems.As quantum computing gains momentum, researchers are increasingly integrating quantum principles with classical metaheuristics like genetic algorithms to address NP-hard challenges such as the traveling salesman problem (TSP), job-shop scheduling, and graph coloring.This review synthesizes peer-reviewed studies published between 2023 and 2025.It highlights key innovations, performance outcomes, convergence behavior, scalability, and hardware constraints.The analysis reveals that GQAs offer significant improvements in solution accuracy, search efficiency, and robustness over classical techniques, though issues such as decoherence, quantum noise, and circuit depth remain open areas of exploration.