= Solution
In heteronuclear <liquid-state NMR quantum computing>, different <gyromagnetic ratios> give well-separated resonance <frequencies>. A resonant transverse radio-frequency pulse selects one nucleus; its <phase> sets the rotation axis in the transverse plane and its duration and <amplitude> set the rotation angle. Detuning, free precession and rotating-frame <phase> changes supply $z$ rotations. For homonuclear <spins>, differing <chemical shifts> can provide <frequency> selectivity, but selective pulses must be narrow enough not to rotate the other <spin>. Exactly magnetically equivalent <spins> cannot be independently addressed by a spatially uniform radio-frequency field merely because they are labelled as different <qubits>.
In the weak-coupling regime, the secular <NMR scalar coupling> is
$$
H_J=2\pi\hbar J I_{1z}I_{2z}=\frac{\pi\hbar J}{2}Z_1Z_2,
$$
with $J$ in hertz. Free coupling evolution for $1/(2|J|)$ gives an entangling propagator $\exp[-i\operatorname{sgn}(J)\pi Z_1Z_2/4]$. Suitable single-qubit $z$ rotations convert it to a <Controlled-Z gate>, and Hadamard conjugation on the target converts controlled-$Z$ to a <controlled-NOT gate>. Refocusing pulses remove unwanted Zeeman evolution and unwanted couplings. This construction applies to resolved, weakly coupled homonuclear <spins> as well as heteronuclear <spins>; strongly coupled or equivalent <spins> need a different control analysis.
These controls supply a universal gate set on a finite addressable connected register, but conventional liquid-state molecular <NMR> is not an architecture that scales straightforwardly to arbitrary register size. A finite chemical-shift bandwidth causes spectral crowding as more <spins> and coupling-split transitions are added; increasingly selective pulses become slow relative to coherence times, and complicated coupling networks require extensive refocusing. The exponentially small pseudo-pure-state signal further limits the complete ensemble scheme. Formal unitary universality therefore does not establish scalable physical addressability, gate speed or initialization.
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