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Exact quantum phase estimation

If

U ∣ ψ ⟩ = e^{2 πij / N} ∣ ψ ⟩

, controlled powers of

U

applied to a uniform

N

-state control register produce

∣ ϕ_{j} ⟩ = \frac{1}{N} \sum_{k = 0}^{N - 1} e^{2 πijk / N} ∣ k ⟩ .

These states are the Fourier basis. Applying the inverse quantum Fourier transform and measuring returns

j

with probability one.

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