Past exam of the laser science and quantum information processing course of the University of Oxford / 2019 / 4 / b / i

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- Solution i

Solution

Apply a Hadamard gate before a Z-basis measurement:

H ∣ + ⟩ = ∣0 ⟩

and

H ∣ - ⟩ = ∣1 ⟩

. The result probabilities are

p_{+} = \frac{∣ α + β ∣ ^{2}}{2}, p_{-} = \frac{∣ α - β ∣ ^{2}}{2} .

For measurement statistics the network is

H

followed by

M_{Z}

. If the measured qubit survives and should be left in the original X eigenstate, append a second

H

after the nondestructive

M_{Z}

. The resulting instrument has Kraus operators

H P_{0} H = ∣ + ⟩ ⟨ + ∣

and

H P_{1} H = ∣ - ⟩ ⟨ - ∣

, not Z-eigenstate outputs.

Figure 1.
Original networks for X measurement, a controlled swap, two-query parity testing, and the simplified Deutsch circuit
. Time runs left to right. Filled controls and circled targets denote computational-basis controlled-NOT operations. In the last network $a = f (0)$ and $b = f (0) \oplus f (1)$ ; powers zero omit a gate. Details.

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