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Cyclic shift diagonalization by the quantum Fourier transform

For the cyclic shift

S ∣ j ⟩ = ∣ j + 1 mod N ⟩

and the convention

QFT_{N} ∣ k ⟩ = N^{- 1/2} \sum_{j} e^{2 πijk / N} ∣ j ⟩,

the Fourier states obey

S QFT_{N} ∣ k ⟩ = e^{- 2 πik / N} QFT_{N} ∣ k ⟩ .

Consequently

S = QFT_{N} D QFT_{N}^{- 1}

, where

D ∣ k ⟩ = e^{- 2 πik / N} ∣ k ⟩

.

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