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
- .
Read more about this topic: Quantum Fourier Transform
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