= Solution
{c}
In <liquid-state NMR quantum computing>, use two distinguishable spin-$1/2$ <atomic nuclei> in one molecule as the two <qubits>. Their different resonance <frequencies> permit selective radiofrequency pulses for one-<qubit> rotations. In a suitable rotating frame, refocusing removes unwanted one-spin terms while the <NMR scalar coupling> supplies
$$
H_J=2\pi\hbar J I_zS_z=\frac{\pi\hbar J}{2}Z_1Z_2.
$$
Assuming $J>0$, free evolution for $t=1/(2J)$ implements $U_J=e^{-i\pi Z_1Z_2/4}$. Define $R_z(\theta)=e^{-i\theta Z/2}$. Then
$$
\boxed{\mathrm{CZ}=e^{-i\pi/4}
[R_z(-\pi/2)\otimes R_z(-\pi/2)]U_J.}
$$
This equality follows directly by evaluating the four diagonal entries: the first three are $+1$ and the $11$ entry is $-1$. Negative $J$ can be handled by refocusing or reversing the effective interaction sign. The <global phase> can be ignored.
Thus the <marked-state phase oracle> for $11$ requires scalar-coupling evolution and calibrated one-spin rotations. For a marked word $b_1b_2$, conjugate by $X_1^{1-b_1}X_2^{1-b_2}$, realized as selective $\pi$ pulses. The <Grover diffusion operator> uses the same entangling operation with additional one-<qubit> pulses. <Liquid-state NMR quantum computing> usually reads an ensemble signal and often prepares an effective pseudo-pure deviation from a large mixed background; that reproduces this small <quantum algorithm>'s signal without providing a scalable source of pure initialized <qubits>.
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