 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-3/10c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Write

x = x_{1} x_{2}

. Since the marked word is

11

, the required oracle is a single Toffoli gate with

x_{1}

and

x_{2}

as its two controls and the answer qubit

y

as its target:

∣ x_{1} x_{2} ⟩ ∣ y ⟩ ⟼ ∣ x_{1} x_{2} ⟩ ∣ y \oplus (x_{1} x_{2})⟩ .

The target flips exactly when both controls are one, which is exactly when

x = 11

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