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Quantum Fourier transform of a periodic coset state

Let

r

divide

N

and

∣ α ⟩ = \frac{1}{N / r} \sum_{j = 0}^{N / r - 1} ∣ x_{0} + j r ⟩ .

The geometric sum in the quantum Fourier transform vanishes unless

c

is a multiple of

N / r

. Thus measurement after the transform returns the

r

values

c = 0, \frac{N}{r}, \dots, (r - 1) \frac{N}{r}

with equal probabilities

1/ r

.

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