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arXiv:1304.3703

Calculation of quantum discord and entanglement measures using the random mutations optimization algorithm

Andrey Yu. Chernyavskiy

Published Apr 12, 2013• 3 citations
arXivHTMLPDF

Abstract

The simple and universal global optimization method based on simplified multipopulation genetic algorithm is presented. The method is applied to quantum information problems. It is compared to the genetic algorithm on standard test functions, and also tested on the calculation of quantum discord and minimal entanglement entropy, which is an entanglement measure for pure multipartite states.

DOI
10.48550/arxiv.1304.3703

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Quantum entanglement which is used as the basis for such applications as quantum cryptography, quantum teleportation, and quantum computing, is a multifaceted property that has attracted much attention. The calculation of quantum entanglement therefore has become important. But the calculation of entanglement has become much more complex as systems that use entanglement have evolved. Based on the relative entropy of entanglement, a general method was developed for the calculation of entanglement for n-qubit or n-qudit system, which using an optimization algorithm technique such as genetic algorithm or Quantum-behaved Particle Swarm Optimization. The method using different optimization algorithm was tested on a two qubit system, for which there are some known points, and compared to exact calculations using the entanglement of formation and with each other. Advantages and disadvantages of the method are discussed and directions of future research are exposed.
Quantum genetic algorithms(QGA) typically construct traditional quantum chromosomes as independent quantum bit systems, which has facilitated the enhancement of search speed and precision in genetic algorithms, demonstrating powerful problem-solving capabilities in optimization issues. Although traditional QGA have enriched population diversity, they still tend to fall into local optima, and the evolutionary update operations require rotation directions to be determined by lookup tables, a process involving multiple conditional judgments that affect algorithm efficiency. To address these issues, an improved method has been proposed to further enhance the performance of QGA. In the initial population selection, a population grouping evolution method based on microhabitat coevolution strategy is used to prevent the algorithm from getting trapped in local optima when solving complex function problems. Additionally, quantum entanglement is introduced on top of quantum super-position by implementing quantum entanglement using quantum registers, and higher-order quantum rotation gates are employed that do not involve multiple conditional judgments of lookup tables, thus improving the efficiency of quantum chromosome measurement and evolutionary updates. Finally, traditional QGA and higher-order quantum genetic algorithms were applied to test functions in multiple experiments. The results indicate that, compared to traditional QGA, the higher-order quantum genetic algorithms have achieved an increase in convergence speed, an approximate 14.3% improvement in convergence precision, and possess enhanced global search capabilities.
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.