Fourier transform

Quantum Fourier Transform (QFT) - Quantum analogy of discrete Fourier transform. For any n-cubic base state | j & # x27E9; {\displaystyle |j\rangle } It works like this: | j & # x27E9; & # x21A6; 1 N ∑ k = 0 N & # x2212; 1 e 2 π i & # xA0; j k / N | k & # x27E9; , {\displaystyle |j\rangle \mapsto {\frac {1}{\sqrt {N}}}\sum _{k=0}^{N-1}e^{2\pi i\ jk/N}|k\rangle ,}

where N = 2 n {\displaystyle N=2^{n}} .

It is worth noting that size ω = e 2 π i / N {\displaystyle \omega =e^{2\pi i/N}} is a "compound of the Nth order" of the number 1 (see de Moivre's formula). This observation helps to visualize how a QFT works by depicting it in the coordinate system of a complex space.

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