 Solution

ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2022/quantum-information/3/b/ii/solution

Since

H = (X + Z) / 2

e^{- iπ (X + Z) / (22)} = - i H .

Thus its Bloch sphere action is a rotation through $π$ about the axis $(1, 0, 1) / 2$ , halfway between the positive

x

and

z

axes:

(r_{x}, r_{y}, r_{z}) ⟼ (r_{z}, - r_{y}, r_{x}) .

The identities

H X H = Z

H Y H = - Y

and

H Z H = X

verify this Bloch vector transformation directly.

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