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The local optima problem is one of the biggest obstacles in compact genetic algorithms since each bit in the problem encoding string is independent of the others. We propose a quantum-assisted compact genetic algorithm that uses a quantum amplitude amplification technique in the selection process to circumvent the said problem. In addition to using elitism mechanics where a single best candidate solution is kept to drive the probability vector, the amplitude amplification subroutine also acts as a mutation operator which, with high probability, enforces the constraint that the newly generated candidate is a feasible solution. We demonstrate this idea by applying the algorithm to the traveling salesman problem of size 3 and 4 cities on IBM Qiskit simulator to show how one would construct the quantum circuit and how to encode the optimization problem into quantum states via Ising spin model encoding. We then show space and time complexity analysis based on the quantum circuit model. Finally, we discuss the number of qubits required for encoding, gate counts in the circuit model, and the practicality of applying this to a small-scale devices in the future.
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