Quantum Fourier Transform

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	CATEGORIES ABOUT QUANTUM FOURIER TRANSFORM
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The quantum Fourier transform has many applications in quantum Algorithm s as it provides the theoretical basis to the ''phase estimation'' procedure. This procedure is the key to quantum algorithms such as Shor's Algorithm for factoring a number, the ''order finding'' algorithm and the ''hidden subgroup'' problem. DETAILS ''l''2(Z/(''N'')) is the Inner Product Space of Complex -valued Function s on Z/''N'' with the inner product

By the discrete Plancherel Theorem , this mapping is a Unitary Transformation .

The main point of the quantum Fourier transform is that in case ''N'' is a power of 2, this operator can be represented as a product. Suppose ''N'' = 2^''n''. We have the orthonormal basis for ''l''²(Z/(''N'')) consisting of the Ket vectors indexed as follows

  :<math> X Angle x_1, x_2, \ldots, x_n angle = x_1 angle \otimes x_2 angle \otimes \cdots \otimes x_n angle</math>