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This paper is the first to propose a novel quantum differential evolutionary algorithm (QDEA) based on the basic quantum-inspired evolutionary algorithm (QEA) for the non-permutation flow-shop scheduling problem (NPFSP). In this QDEA, the quantum chromosomes are encoded by using the quantum rotating angle and a simple converting mechanism for determining job sequence is proposed for the representation of NPFSP firstly. Then we merge the advantages of differential operation, local search and QEA by adopting the differential operation to perform the updating of quantum gate and the local search to perform exploitation in the promising permutative-based solutions. We adopt this approach to minimize the makespan for NPFSP and make the simulation. The comparisons with other state-of-the-art approaches based on well-known benchmarks demonstrate the effectiveness of the proposed QDEA.