 Solution

ID: past-exam-of-the-introduction-to-quantum-information-course-of-the-university-of-oxford/2021/1/c/solution

(X, Y, Z)

(Z, - Y, X)

. Therefore its Bloch vector action is

H : (s_{x}, s_{y}, s_{z}) ⟼ (s_{z}, - s_{y}, s_{x}) .

It is a rotation through

π

about

(e_{x} + e_{z}) / 2

H = (X + Z) / 2

differs by a global phase from the corresponding spin rotation.

The S gate and T gate differ by global phases from

z

S : R_{z} (π /2), T : R_{z} (π /4) .

Indeed

diag (1, e^{i θ}) = e^{i θ /2} e^{- i θZ /2}

, which sends

(s_{x}, s_{y}, s_{z})

(s_{x} cos θ - s_{y} sin θ, s_{x} sin θ + s_{y} cos θ, s_{z})

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