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Based on the analysis of the stochastic earliness and tardiness parallel machine scheduling, a Quantum Genetic Scheduling Algorithm (QGSA) is presented. In the QGSA, the Q-bit based representation in discrete 0-1 hyperspace is employed, which is then converted into decimal scheduling code and quantum gate is used to update the current generation, meanwhile catastrophe operator is added to avoid premature. In contrast to the deterministic scheduling model where jobs with their processing time are known beforehand, we propose a stochastic expected value model based on stochastic programming theory. The simulation results demonstrate that QGSA can get better scheduling solution, even with a small population, without premature convergence as compared to the conventional Genetic Algorithm (GA).