Quantum Fourier transform: Difference between revisions

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== Definition ==

The quantum Fourier transform is the classical discrete Fourier transform applied to the vector of amplitudes of a quantum state. The classical (unitary) Fourier transform acts on a [[vector (mathematics and physics)|vector]] in <math>\mathbb{C}^N</math>, (''x''₀, ..., ''x''_''N''−1) in <math>\mathbb{C}^N</math> and maps it to the vector (''y''₀, ..., ''y''_''N''−1) according to the formula:

:<math>y_k = \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} x_j \omega^{jk}</math>,

where <math>\omega^{jk} = e^{2 \pi i\frac{j\, k}{N}}</math> is a ''N''^th [[root of unity]].