 Solution

ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2019/quantum-information/1/b/i/solution

The columns are

i ∣ + ⟩

and

- i ∣ - ⟩

. Their norms are one and their inner product is zero, since

∣ + ⟩, ∣ - ⟩

are orthonormal. Thus $M_{1}$ is a unitary matrix. Equivalently,

M_{1} = i H Z, M_{1}^{†} M_{1} = Z H^{†} H Z = I,

using the Hadamard gate and Pauli Z gate. The different column phases do not destroy orthonormality.

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