Quantum Fourier Transform - Definition

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 in, (x0, ..., xN−1) and maps it to the vector (y0, ..., yN−1) according to the formula:

where is a primitive Nth root of unity.

Similarly, the quantum Fourier transform acts on a quantum state and maps it to a quantum state according to the formula:

.

This can also be expressed as the map

.

Equivalently, the quantum Fourier transform can be viewed as a unitary matrix acting on quantum state vectors, where the unitary matrix is given by

F_N = \frac{1}{\sqrt{N}} \begin{bmatrix} 1&1&1&1&\cdots &1 \\ 1&\omega&\omega^2&\omega^3&\cdots&\omega^{N-1} \\ 1&\omega^2&\omega^4&\omega^6&\cdots&\omega^{2(N-1)}\\ 1&\omega^3&\omega^6&\omega^9&\cdots&\omega^{3(N-1)}\\ \vdots&\vdots&\vdots&\vdots&&\vdots\\ 1&\omega^{N-1}&\omega^{2(N-1)}&\omega^{3(N-1)}&\cdots&\omega^{(N-1)(N-1)}\\ \end{bmatrix} .

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