A Gate-Based Quantum Genetic Algorithm for Real-Valued Global Optimization

TL;DR

This work introduces a gate-based Quantum Genetic Algorithm (QGA) for real-valued global optimization, where each candidate is a quantum circuit whose measurement statistics are decoded into real vectors via binary encoding. The model supports fixed-depth and variable-depth circuit variants, with evolutionary operators (mutation, crossover) acting on circuit structure and gate content; fitness is estimated by quantum sampling of the circuit outputs. By comparing gate sets with and without Hadamard gates, the authors show that quantum superposition enhances convergence and robustness on benchmark functions, and they further demonstrate that inter-individual entanglement accelerates early convergence, indicating a genuine quantum advantage at both the intra- and inter-circuit levels. The results suggest that gate-based QGAs can harness quantum resources for improved global optimization and motivate future hardware implementations and extensions to more complex domains.

Abstract

We propose a gate-based Quantum Genetic Algorithm (QGA) for real-valued global optimization. In this model, individuals are represented by quantum circuits whose measurement outcomes are decoded into real-valued vectors through binary discretization. Evolutionary operators act directly on circuit structures, allowing mutation and crossover to explore the space of gate-based encodings. Both fixed-depth and variable-depth variants are introduced, enabling either uniform circuit complexity or adaptive structural evolution. Fitness is evaluated through quantum sampling, using the mean decoded output of measurement outcomes as the argument of the objective function. To isolate the impact of quantum resources, we compare gate sets with and without the Hadamard gate, showing that superposition consistently improves convergence and robustness across benchmark functions such as the Rastrigin function. Furthermore, we demonstrate that introducing pairwise inter-individual entanglement in the population accelerates early convergence, revealing that quantum correlations among individuals provide an additional optimization advantage. Together, these results show that both superposition and entanglement enhance the search dynamics of evolutionary quantum algorithms, establishing gate-based QGAs as a promising framework for quantum-enhanced global optimization.

Figures (7)

• Figure 1: Best fitness obtained on the Rastrigin function as a function of circuit depth for a population size of $|\mathcal{P}|=50$ and different numbers of generations ($G_{\max}=10, 30, 50$). Results are shown for both the classical gate set (red) and the quantum gate set including superposition (blue). Each curve represents the average over 5000 independent repetitions.
• Figure 2: Fitness of the best individuals as a function of the Shannon entropy of the circuits for the Rastrigin function with population size $|\mathcal{P}|=50$ and $G_{\max}=50$. The classical configuration corresponds to the zero-entropy case (red), while the quantum configuration explores a broad entropy spectrum (blue).
• Figure 3: Log–log plot of the fitness of the best individuals as a function of the number of generations for the Rastrigin function with population size $|\mathcal{P}|=50$. Results are shown for both the classical (red) and quantum (blue) configurations, each obtained by averaging over 5000 independent runs.
• Figure 4: Circuit depth distribution after 140 generations for the Rastrigin function with population size $|\mathcal{P}|=50$. Both the classical (red) and quantum (blue) configurations favor deeper circuits, reflecting the evolutionary tendency toward greater expressiveness. Results are averaged over 5000 runs.
• Figure 5: Fitness versus Shannon entropy after 150 generations for the Rastrigin function with population size $|\mathcal{P}|=50$. The classical configuration (red) remains at zero entropy, while the quantum configuration (blue) spans a broad range of values, with higher entropy correlating with improved fitness. Results are averaged over 5000 runs.