Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 10C / b

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- Solution b

Solution

For every product vector,

(A \otimes B) (∣ x ⟩ \otimes ∣ y ⟩) = A ∣ x ⟩ \otimes B ∣ y ⟩ .

Product vectors span the two-register state space, so the single

A \otimes B

gate equals parallel application of

A

to the first register and

B

to the second. Unitarity guarantees that both descriptions are valid quantum circuits.

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