The universality of the quantum Fourier transform in forming the basis of quantum computing algorithms

The quantum Fourier transform (QFT) is a powerful tool in quantum computing. The main ingredients of QFT are formed by theWalsh–Hadamard transform H and phase shifts P(·), both of which are 2×2 unitary matrices as operators on the two-dimensional 1-qubit space. In this paper, we show that H and P(·) suffice to generate the unitary group U(2) and, consequently, through controlled-U operations and their concatenations, the entire unitary group U(2n) on n qubits can be generated. Since any quantum computing algorithm in an n-qubit quantum computer is based on operations by matrices in U(2n), in this sense we have the universality of the QFT.  2002 Elsevier Science (USA). All rights reserved.