  The quantum Fourier transform (QFT) is a powerful tool in quantum computing.
The main ingredients of QFT are formed by the Walsh-Hadamard transform H and
phase shifts P(.), both of which are 2x2 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(2^n) 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(2^n), in this sense we
have the universality of the QFT.

Quantum Physics