 Solution

The Hadamard gate H and T gate $T=\mathrm{diag}(1,e^{i\pi /4})$ form a universal quantum gate set for single-qubit operations: finite sequences approximate any target single-qubit unitary arbitrarily well, up to a physically irrelevant global phase. Universality is not a claim that every arbitrary rotation has a finite exact H,T word. With products acting right to left, exact Pauli constructions areIndeed $T^2=S$, $T^6=S^{†}$, and $SX{S}^{†}=Y$.

Initialize the search register in $|00⟩$ and the output qubit in $|1⟩$, then apply H to all three. The search state becomes $|s⟩=(|00⟩+|01⟩+|10⟩+|11⟩)/2$, and the output becomes $|-⟩$. The oracle obeysThis phase kickback leaves the output qubit factored out, so subsequent analysis can use the phase oracle $O_f$ on the two search qubits alone. Apply one Grover diffusion operator $D=2|s⟩⟨s|-I$, then measure the search register.

A promise restricts which Boolean functions may be supplied. For the exact one-iteration four-entry search here, promise exactly one marked input. The four functions are $f_{jk}(x)=1$ precisely when $x=jk$, for $jk\in \{00,01,10,11\}$. Other known numbers of marked entries require a correspondingly chosen iteration count; uniqueness is the promise for this particular exact circuit.

In basis order $00,01,10,11$, the requested explicit calculation isThus the promised marked entry is found with probability one.

For two weakly coupled spin-half nuclei in nuclear magnetic resonance, a useful rotating-frame Hamiltonian iswhere $J$ is in hertz and $I_z=Z_1/2$, $S_z=Z_2/2$. Resonant pulses and refocusing separate local rotations from the scalar coupling. For positive $J$, evolution for $t=1/(2J)$ supplies $e^{-i\pi Z_1{Z}_2/4}$. Combining it with local Z phases gives the phase oracleThe scalar global phase is immaterial. Local $\pi$ pulses interchange zero and one, soThis implements the desired change of marked input without an additional output qubit.