Past exam of the computer science course of the University of Oxford / 2021 / Quantum information / 3 / a

Prepare $|0⟩$, then apply $H$, then $T$. With $T=\mathrm{diag}(1,e^{i\pi /4})$,This uses only the allowed state preparation and elementary gates. The rightmost operator acts first, so $TH$ denotes the temporal order $H$ followed by $T$.

The required superposition is the Bell statePrepare $|00⟩$, apply $H$ to the first qubit, then apply CNOT from the first to the second. The intermediate state is $(|00⟩+|10⟩)/\sqrt{2}$, and the controlled-NOT gate gives the required Bell state. No final basis change is needed: the two displayed expressions represent exactly the same vector.