Quantum circuit design of discrete Hartley transform using recursive decomposition formula

In this paper, the quantum circuit design of a discrete Hartley transform (DHT) is investigated. The proposed design procedure can be divided into the following two steps. First, the N-point transform matrix is decomposed into the combination of N/2-point transform matrices and several sparse matrices such that any 2ⁿ-point transform can be computed recursively from the 2-point transform. Second, starting from the 2-point Hadamard transform, the quantum circuit of 2ⁿ-point transform can be designed recursively by using elementary quantum gates. The design results are compared with those of the conventional factorization method to demonstrate the effectiveness of proposed method.