The Universality of the Quantum Fourier Transform in Forming the Basis of Quantum Computing Algorithms
Charles M. Bowden, Goong Chen, Zijian Diao, Andreas Klappenecker
Abstract
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(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.
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