MATLAB Implementation of Quantum Genetic Algorithm

The Quantum Genetic Algorithm (QGA) is an optimization method that combines quantum computing principles with traditional genetic algorithms, primarily used to solve complex optimization problems. Unlike conventional genetic algorithms, QGA leverages the superposition properties of quantum bits (qubits), enabling populations to maintain higher diversity and search efficiency during evolution.

### Core Algorithm Concepts Qubit Encoding: Unlike traditional binary encoding in genetic algorithms, QGA uses quantum bits to represent chromosome states. A qubit can be expressed as a probability amplitude combination, such as |ψ⟩ = α|0⟩ + β|1⟩, where |α|² + |β|² = 1. This representation allows the population to maintain multiple state possibilities simultaneously. Quantum Rotation Gate Adjustment: The probability amplitudes of qubits are updated through the quantum rotation gate, guiding the population toward better solutions. The rotation angle is typically determined by the current individual's fitness. Observation and Classical Genetic Operations: During each generation, qubits are collapsed into definitive binary encodings (0 or 1) through observation, followed by genetic operations like crossover and mutation. Fitness Evaluation: Calculate the fitness of each individual in the current population and select the best individuals for the next generation.

### MATLAB Implementation Approach Initialization of Quantum Population: Randomly generate a set of qubit chromosomes, with initial α and β values typically set to √0.5 to ensure uniform initial distribution. Iterative Optimization Process: - Observation to Generate Classical Population: Convert qubits to binary encoding through quantum state collapse. - Fitness Calculation: Evaluate fitness values for each individual and record the optimal solution. - Quantum Rotation Gate Update: Adjust qubit probability amplitudes based on fitness and current best solution. - Genetic Operations: Perform crossover and mutation on the classical population to enhance diversity. Termination Condition: Usually based on maximum iterations or fitness convergence.

### Implementation Details Key MATLAB functions involved: - qubit_initialization(): Creates initial population with qubit representation using rand() function for probability amplitudes - observe_qubits(): Converts qubits to binary strings by comparing probability amplitudes with random thresholds - quantum_rotation_gate(): Implements rotation matrix operations using angle parameters derived from fitness differences - fitness_evaluation(): Custom function specific to the optimization problem, returning scalar fitness values

### Extended Applications Quantum genetic algorithm shows promising applications in NP-hard problems, machine learning parameter optimization, and path planning. Compared to traditional genetic algorithms, it more effectively escapes local optima and suits high-dimensional complex optimization problems. The MATLAB implementation typically requires 50-200 lines of code depending on problem complexity, with main computational overhead in quantum gate operations and fitness evaluation.