Past exam of the laser science and quantum information processing course of the University of Oxford / 2022 / 4 / c / Solution

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 suppliesAssuming $J>0$, free evolution for $t=1/(2J)$ implements $U_J=e^{-i\pi Z_1{Z}_2/4}$. Define $R_z(\theta )=e^{-i\theta Z/2}$. ThenThis 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 $b_1{b}_2$, 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.