Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 324 / 3 / b

Table of contents
- Solution b

Solution

The shift

S ∣ x ⟩ = ∣ x - 1 mod Q ⟩

acts on a Fourier state as

S QFT_{Q} ∣ a ⟩ = \frac{1}{Q} x \sum ω^{a x} ∣ x - 1 ⟩ = ω^{a} QFT_{Q} ∣ a ⟩ .

Thus

QFT_{Q} ∣ a ⟩

is an eigenvector of

S

with eigenphase

a / Q

. Apply the unitary part of exact quantum phase estimation for

S

to a zeroed control register and this Fourier state. It writes the eigenphase label coherently:

∣ 0^{m} ⟩ QFT_{Q} ∣ a ⟩ ⟼ ∣ a ⟩ QFT_{Q} ∣ a ⟩ .

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