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In this paper, the quantum circuit design of an 8×8 discrete cosine transform (DCT) is investigated. The proposed design procedure can be divided into the following three steps. First, the DCT matrix is decomposed into the product of sparse matrices, based on its fast computation flow graph. Second, each sparse matrix is implemented by elementary quantum gates. Third, the sparse matrix circuits are cascaded to obtain the final circuit. The proposed method makes the fast DCT computation algorithm in digital signal processing suitable for implementation in a quantum computer.