Past exam of the computer science course of the University of Oxford / 2020 / Quantum information / 3 / b / iii

Table of contents
- Solution iii

Solution

For the marked subset,

⟨ e_{0} ∣ A_{1} ⟩ = S_{1} / N

by the same summation of the individual inner products. Consequently

W ∣ A_{1} ⟩ = 2∣ e_{0} ⟩ \frac{S _{1}}{N} - ∣ A_{1} ⟩ = \frac{2 S _{1}}{N} ∣ J ⟩ - ∣ A_{1} ⟩ .

The cardinality is

S_{1}

here. Its sign will reverse when this expression is used for the oracle's negative marked component.

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