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Duality quantum computing is a new mode of a quantum computer to simulate a moving quantum computer passing through a multi-slit.It exploits the particle wave duality property for computing.A quantum computer with n qubits and a qudit simulates a moving quantum computer with n qubits passing through a d-slit.Duality quantum computing can realize an arbitrary sum of unitaries and therefore a general quantum operator,which is called a generalized quantum gate.