The Universality of the Quantum Fourier Transform in Forming the Basis of Quantum Computing Algorithms
Quantum Physics
2007-05-23 v1
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(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.
Cite
@article{arxiv.quant-ph/0007122,
title = {The Universality of the Quantum Fourier Transform in Forming the Basis of Quantum Computing Algorithms},
author = {Charles M. Bowden and Goong Chen and Zijian Diao and Andreas Klappenecker},
journal= {arXiv preprint arXiv:quant-ph/0007122},
year = {2007}
}
Comments
12 pages, 3 figures, submitted to SIAM Journal on Computing